Equations: Difference between revisions

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Forming Equations

Before proceeding to the actual solution of an equation of any type, certain preliminary operations have necessarily to be carried out in order to prepare it for solution.

Still more preliminary work is that of forming the equation ( samī-karaṇa, samī-kāra or samī-kriyā; from sama, equal and kṛ , to do; hence literally , making equal) from the conditions of the proposed problem. Such preliminary work may require the application of one or more fundamental operations of algebra or arithmetic.

Bhāskara II says: "Let yāvat-tāvat be assumed as the value of the unknown quantity. Then doing precisely as has been specifically told-by subtracting, adding, multiplying or dividing the two equal sides of an equation should be very carefully built.

Algebraic Expressions and Algebraic Equations

Algebraic expression can be understood with the following example.

Ram says that he has 10 coins more than Shyam. We do not know exactly how many coins Ram has. Ram may have any number of coins. With the given information

Number of coins held by Ram = Number of coins held by Shyam + 10

We will denote the ‘number of coins held by Shyam by the letter x. Here x is unknown which could be 1, 2, 3, 4, etc.

Using x, we write,

Number of coin held by Ram = x+10.

Thus 'x + 10' is an algebraic expression.

Algebra utilizes the usage of symbols. These symbols represent the unknown quantities and operations performed with them. The following table gives the symbols which were used for some basic operations by the ancient Indian Mathematicians.


S. No.	Components of an algebraic expression	Samskrit word	Symbol	Examples
1	Unknown	यावत्तावत् कालकः नीलकः , ......	या का नी , ........	या ३५ का १४ नी ८२	35x 14y 82z
2	Sum	योगः	-	या का या ३५ का १४	x + y 35x + 14y
3	Product	भावितम्	भा	याकाभा याकाभा ३२	xy 32xy
4	Square	वर्गः	व	याव	x²
5	Cube	घनः	घ	याघ	x³
6	Fourth Power	वर्ग-वर्गः	वव	यावव	x⁴
7	Constant Term	रूपम्	रू	रू ३२	32
8	Negative	ऋणम्	dot above the quantity (.)	. रू ४३२	-432

The letter yā (an abbreviation of yāvat-tāvat) was the most popular representation of the unknown quantity. Its square was termed yāva, the abbreviation of yāvat-tāvat-varga (varga means square). constant term was denoted by the letter rū, an abbreviation of rūpa as shown in the above table. Any negative sign in the equation is denoted by a dot above the term.

If there are three unknown quantities in an expression, the symbols used are yā, kā and nī. These are the abbreviations for yāvat tāvat, kālaka and nīlaka. The product of first two unknown quantities is represented as yākābha where yā and kā are the two unknowns and bha stands for their product.

The following table gives a representation of some of the algebraic expressions used by ancient Indian mathematicians.


S.No.	Modern Notation	Ancient Indian Notation
1	x + 17	या १ रू १७
2	7x - 17	या ७ रू १७^.
3	18x – 8	या १८ रू ८^.
4	15x² + 17x - 2	याव १५ या १७ रू २^.
5	1x⁴ + 16x³ + 25x² + 8x + 6	यावव १ याघ १६ याव २५ या ८ रू ६
6	8x² + 12xy - 6xz -16x	याव ८ याकाभा १२ यानीभा ६^. या १६^.

How algebraic expressions are written by Ancient Indian mathematicians.

Consider the equation 10 x - 18 = x² +14

This can be written as,

0x² + 10 x - 18 = 1x² + 0x + 14

By looking at the positions of x², x¹, x⁰ (constant term), there is some pattern ? The standard way of writing an equation starts from the highest power of x. Then the powers of x were written in the descending order up to its lowest power. This format of writing equation was followed by mathematicians from ancient times.

Brahmagupta called an equation as samakaraṇa or samīkaraṇa. It means 'making equal. The two sides of an equation (LHS and RHS) were written one below the other. The symbol '=' was not used. Both the sides of an equation were made same by finding the appropriate value(s) for the unknown(s).

Pṛthūdakasvāmin (864 CE) in his commentary on the Brāhma-sphuṭa-siddhānta writes the equation 40x - 48 = x² + 51 as below


Devanāgari	Transliteration		Modern notation
याव ० या ४० रू ४८^. याव १ या ० रू ५१	Yāva 0 yā 40 rū 48^. Yāva 1 yā 0 rū 51	⇒	0x² + 40 x - 48 = 1x² + 0x + 51

Here is an example of an equation from Bījagaṇita of Bhāskara II is:

X⁴ - 2x² - 400x = 9999

This is represented as,

यावव १ याव २^. या ४^.०० रू ०

यावव ० याव ० या ० रू ९९९९

Operations with Algebraic Expressions

Bhāskara II gives the operations using algebraic terms as follows:

स्याद्रूपवर्णाभिहतौ तु वर्णो द्वित्र्यादिकानां समजातिकानाम् ॥

वधे तु तद्वर्गघनादयः स्युस्तद्भावितं चासमजातिघाते।

भागादिकं रूपवदेव शेषं व्यक्ते यदुक्तं गणिते तदत्र ॥^[1]

“The product of a numerical constant and an unknown quantity is an unknown quantity. Products of two or three like terms are their squares or cubes (respectively). Product of unlike terms is bhāvita. Fractions etc. are as in the case of knowns. The other (processes) are same as explained in arithmetic."

Addition and Subtraction of Algebraic Expressions

Bhāskara II gives the rule for addition and subtraction of unknown quantities as follows:

योगोऽन्तरं तेषु समानजात्योर्विभिन्नजात्योश्च पृथक् स्थितिश्च।^[2]

“Addition and subtraction are performed amongst like terms. The unlike terms are to be kept separately."

Explanation:

Addition and subtraction can be performed with like terms and unlike terms are to be kept separately. Same letter variables raised to the same powers are treated as like terms. E.g., या ४, या ५, या ६ are like terms. याव ७, याव ८, याव ९ are also like terms. का ३, का ७, का १५ are also like terms. Presently we say 4x, 5x, 6x are like terms. Similarly 7x², 8x², 9x² are like terms. and 3y, 7y, 15y are also like terms. When we have like terms, the sum and difference can be simplified. E.g. 4x + 6x can be simplified as 10x. 9x² - 7x² can be simplified as 2x².

Unlike terms are those terms having different variables or variables with different powers. E.g. या ३, याव ३, याघ ४, का ५, काव, याकाभा . Presently, these are represented as 3x, 3x², 4x³, 5y, y², xy.

Multiplication of Algebraic Expressions

Bījagaṇita gives a rule for multiplication -

गुण्यः पृथग्गुणकखण्डसमो निवेश्यस्तैः खण्डकैः क्रमहतः सहितो यथोक्त्या।

अव्यक्तवर्गकरणीगणनास चिन्त्यो व्यक्तोक्तखण्डगुणनाविधिरेवमत्र॥^[3]

“Place the multiplicand at as many places as the terms of the multiplier. Multiply with the terms of the multiplier in order separately and add the results as directed in the problem. This is applicable in the case of squares of unknown numbers and surds also. The method of partial products stated in the case of arithmetic numbers is applicable here also.”

Explanation


Ancient Indian notation	Modern notation
If या २ रू ४ and या ३ रू ५ are multiplicand and multiplier respectively, we can get their product as mentioned below	If 2x + 4 and 3x + 5 are multiplicand and multiplier respectively, we can get their product as mentioned below
The multiplier contains two terms, i.e., या ३ and रू ५	The multiplier contains two terms, i.e., 3x and 5
Multiply the multiplicand with the two terms of the multiplier separately as mentioned below. (या २ रू ४) X या ३ = याव ६ या १२ (या २ रू ४) X रू ५ = या १० रू २०	Multiply the multiplicand with the two terms of the multiplier separately as mentioned below. (2x + 4) X 3x = 6x² + 12x (2x + 4) X 5 = 10x + 20
Add the results. The result of the multiplication is : याव् ६ या २२ रू २०	Add the results. The result of the multiplication is : 6x² + 22x + 20

If $ax+b$ and $cx+d$ are multiplicand and multiplier respectively, we get their product as mentioned below.

The multiplier has two terms, i.e., cx and d. Multiply the multiplicand with the two terms of the multiplier separately as mentioned below.

$(ax+b)cx=acx^{2}+bcx$

$(ax+b)d=adx+bd$

Add the results.

The multiplication result is : $acx^{2}+(bc+ad)x+bd$

Algebraic Notations

The symbols used for unknown numbers are the initial syllables yа̄ of yа̄vat-tа̄vat (as much as), kа̄ of kа̄laka (black), nī of nīlaka (blue), pī of pīta (yellow) etc.

The product of two unknowns is denoted by the initial syllable bhā of bhāvita (product) placed after them. The powers are denoted by the initial letters va of varga (square), gha of ghana (cube); vava stands for vargavarga, the fourth power. Sometimes the initial syllable ghā of ghāta (product) stands for the sum of powers.

A coefficient is placed next to the symbol. The constant term is denoted by the initial symbol rū of rūpa (form).

A dot is placed above the negative integer

The two sides of an equation are placed one below the other. Thus the equation X⁴ - 2X² - 400x = 9999; is written as

यावव १ याव २^● या ४००^● रू ०

यावव ० याव ० या ० रू ९९९९

which means writing x for या

x⁴ -2x² -400x+0 = 0x⁴ +0x²+0x+9999

If there be several unknowns, those of the same kind are written in the same column with zero coefficients, if necessary. Thus the equation

197x - 1644y - z = 6302 is represented by

yа̄ 197 kа̄ 1644^● ni 1^● ru 0

yа̄ 0 kа̄ 0 ni 0 ru 6302

which means, putting y for kа̄ and z for ni

197x - 1644Y - z + 0 = 0x + 0y + 0z + 6302.

Bhāskara II says:

"Then the unknown on one side of it (the equation) should be subtracted from the unknown on the other side; so also the square and other powers of the unknown;

the known quantities on the other side should be subtracted from the known quantities of another side."

The following illustration is from the Bījagaṇita of Bhāskara II:

"Thus the two sides are

yā va 4 yā 34^● rū 72

yā va 0 yā 0 rū 90

On complete clearance (samaśodhana), the residues of the two sides are

yā va 4 yā 34^● rū 0

yā va 0 yā 0 rū 18

i.e., 4x² - 34x = 18

Classification of Equations

The earliest Hindu classification of equations seems to have been according to their degrees, such as simple (technically called yāvat tāvat ), quadratic (varga), cubic (ghana) and biquadratic (varga-varga). Reference to it is found in a canonical work of circa 300 B.C. But in the absence of further corroborative evidence, we cannot be sure of it. Brahmagupta (628) has classified equations as: (I) equations in one unknown (eka-varna-samīkaraṇa), (2) equations in several unknowns (aneka-varna-samīkaraṇa), and (3) equations involving products of unknowns (bhaivita).

The first class is again divided into two sub classes, viz.,(i) linear equations, and (ii) quadratic equations (avyakta-varga-samīkaraṇa). Here then we have the beginning of our present method of classifying equations according to their degrees. The method of classification adopted by Pṛthūdakasvāmī (860) is slightly different. His four classes are: (1) linear equations with one unknown, (2) linear equations with more unknowns, (3) equations with one, two or more unknowns in their second and higher powers, and (4) equations involving products of unknowns. As the method of solution of an equation of the third class is based upon the principle of the elimination of the middle term, that class is called by the name madhyamāharaṇa (from madhyama, "middle", aharana "elimination", hence meaning "elimination of the middle term"). For the other classes, the old names given by Brahmagupta have been retained. This method of classification has been followed by subsequent writers.

Bhāskara II distinguishes two types in the third class, viZ" (i) equations in one unknown in its second and higher powers and (ii) equations having two or more unknowns in their second and higher powers.' According to Krsna (1580) equations are primarily of two classes: (1) equations in one unknown and (z) equations in two or more unknowns. The class (1), again, comprises two subclasses: (i) simple equations and (ii) quadratic and higher equations. The class (2) has three subclasses: (i) simultaneous linear equations, (ii) equations involving the second and higher powers of unknowns, and (iii) equations involving products of unknowns. He then observes that these five classes can be reduced to four by including the second subclasses of classes (1) and (2) into one class as madhyamāharaṇa.

Linear Equations in One Unknown

A Linear equation is an equation containing only the first power of the variables, coefficients and constants. For example, the equation 2x + 4 = 5 is a linear equation in one variable. This is called first-order equation since the power of the variable (x) is one. If the equation has highest power of x as two, i.e. x² , then it will be a quadratic (second order) equation.

Early Solutions:

As already stated, the geometrical solution of a linear equation in one unknown is found in the śulba , the earliest of which is not later than 800 B.C. There is a reference in the Sthānāṅga-Sūtra (c. 300 B.C.) to a linear equation by its name (yāvat -tāvat ) which is suggestive of the method of solution! followed at that time.We have, however, no further evidence about it. The earliest Hindu record of doubtless value of problems involving simple algebraic equations and of a method for their solution occurs in the Bakhshālī treatise, which was written very probably about the beginning of the Christian Era.

One problem is "The amount given to the first is not known. The second is given twice as much as the first; the third thrice as much as the second; and the fourth four times as much as the third. The total amount distributed is 132. What is the amount of the first?"

If x be the amount given to the first, then according to the problem,

x + 2X + 6x + 24X = 132.

Rule of False Position:

The solution of this equation is given as follows:

" 'Putting any desired quantity in the vacant place' ; any desired quantity is 1 ; 'then construct the series.


1	2	2 3	6 4
1	1	1 1	1 1

'multiplied'


1	2	6	24

added' 33. "Divide the visible quantity'


132 33

(which) on reduction becomes


4 1

(This is) the amount given (to the first)."

The solution of another set of problems in the Bakhshālī treatise, leads ultimately to an equation of the type ax+ b=p. The method given for its solution is to put any arbitrary value g for x, so that

ag+ b =p' say.

Then the correct value will be

${\displaystyle x={\frac {(p-p')}{a}}+g}$

Solution of Linear Equations

āryabhaṭaI(499) says:

"The difference of the known "amounts" relating to the two persons should be divided by the difference of the coefficients of the unknown. The quotient will be the value of the unknown, if their possessions be equal."

This rule contemplates a problem of this kind: Two persons, who are equally rich, possess respectively a, b times a certain unknown amount together with c, d

units of money in cash. What is that amount?

If x be the unknown amount, then by the problem

ax+ c= bx+ d.

Therefore ${\displaystyle x={\frac {(d-c)}{(a-b)}}}$

Hence the rule.

Rule for solving the linear equation of the form bx + c = dx + e where b, c, d and e are given numbers is given by Brahmagupta as follows.

अव्यक्तान्तरभक्तं व्यस्तं रूपान्तरं समेऽव्यक्तः।

वर्गाव्यक्ताः शोध्या यस्माद्रूपाणि तदधस्तात् ॥^[4]

"The difference of absolute numbers, inverted and divided by the difference of unknown, is the [value of the) unknown in an equation.”

Explanation: Consider the equation, bx + c = dx + e

Here x is the unknown quantity whose value is to be found. The letters b and d are its coefficients. The remaining letters c and e are numerical constants.

Difference of absolute numbers = c-e

Difference of absolute numbers inverted = e-c

Difference of coefficients of unknown = b - d

x is found as

${\displaystyle x={\frac {(e-c)}{(b-d)}}}$

Bhāskara II explains how the above formula is obtained.

यावत्तावत् कल्प्यमव्यक्तराशेर्मानं तस्मिन् कुर्वतोद्दिष्टमेव ।

तुल्यौ पक्षौ साधनीयौ प्रयत्नात्त्यक्त्वा क्षिप्त्वा वाऽपि संगुण्य भक्त्�

[1]

[2]

[3]

[4]

@@ Line 709: / Line 709: @@
 == Quadratic Equations ==
-The geometrical solution of the simple quadratic equation
+In the early canonical works of the Jainas (500-300 B.C) we see the  geometrical solution of the simple quadratic equation
-<math>4h^2 -4dh =- c^2</math> is found in the early canonical works of the Jainas (500- 300 B. C.) and also in the ''Tattvathadhigama-Sūtra''  of Umasvati (c. 150 B. C.) as .
+<math>4h^2 -4dh =- c^2</math> . Also in the ''Tattvathadhigama-Sūtra''  of Umasvati (c. 150 B. C.) as  <math>{\displaystyle h = {\frac {1}{2}} (d-\sqrt{d^2-c^2})}</math>.
-<math>{\displaystyle h = {\frac {1}{2}} (d-\sqrt{d^2-c^2})}</math>
+'''śrīdhara's Rule.'''   śrīdhara (c. 750) clearly indicates his method of solving the quadratic equation.
-'''Sridhara's Rule.''' Sddhara (c. 750) expressly indicates his method of solving the quadratic equation.
 His treatise on algebra is now lost. But the relevant portion of it is preserved in quotations by Bhāskara II and others. Sridhara's method is: .
@@ Line 723: / Line 721: @@
 quantity equal to the square of the (original) coefficient of the unknown: then extract the root."
-That is, to solve
+To solve the equation
 <math>ax^2 + bx = c</math>
-we have <math>4a^2x^2 + 4abx = 4ac</math>
+Multiply by 4a on both sides
-or
+<math>4a^2x^2 + 4abx = 4ac</math>
 <math>(2ax+b)^2 = 4ac + b^2</math>
 <math>2ax+b = \sqrt{4ac + b^2}</math>
@@ Line 737: / Line 735: @@
 <math>x= \frac{\sqrt{4ac+b^2} -b}{2a}</math>
-'''Sripati's Rules'''. Sripati (1039) indicates two methods of solving the quadratic. There is a lacuna in our manuscript in the rule describing the first method, but it can be easily recognized to be the same as that of Sridhara. "Multiply by four times the coefficient of the square of the unknown and add the square of the coefficient of the unknown; then extract the square-root divided by twice the coefficient of the square of the unknown, is said to be the value of the unknown." "Or multiplying by the coefficient of the square of the unknown and adding the square of half the coefficient of the unknown, extract the square-root. Then proceeding as before, it is diminished by half the coefficient of the unknown and divided by the coefficient of the square of the unknown. This quotient is said to be the value of the unknown."
+'''Sripati's Rules'''. Sripati (1039) indicates two methods of solving the quadratic. There is a lacuna in our manuscript in the rule describing the first method, but it can be easily recognized to be the same as that of  śrīdhara .
+"Multiply by four times the coefficient of the square of the unknown and add the square of the coefficient of the unknown; then extract the square-root divided by twice the coefficient of the square of the unknown, is said to be the value of the unknown."
+"Or multiplying by the coefficient of the square of the unknown and adding the square of half the coefficient of the unknown, extract the square-root. Then proceeding as before, it is diminished by half the coefficient of the unknown and divided by the coefficient of the square of the unknown. This quotient is said to be the value of the unknown."
 <math>ax^2 + bx = c</math>
@@ Line 748: / Line 750: @@
 <math>{\displaystyle ax+ \left ( \frac{b}{2} \right ) = \sqrt{ac + \left ( \frac{b}{2} \right ) ^2} }
 </math>
@@ Line 753: / Line 756: @@
 </math>
-'''Bhāskara II's Rules.''' Bhāskara- II (1150) says: "When the square of the unknown, etc., remain, then, multiplying the two sides (of the equation) by some suitable quantities, other suitable quantities should be added to them so that the side containing the unknown becomes capable of yielding a root (''pada-prada''). The equation should then be formed again with the root of this side and the root of the known side. Thus the value of the unknown is obtained from that equation.
+'''Bhāskara II's Rules.'''  Bhāskara- II (1150) says: "When the square of the unknown, etc., remain, then, multiplying the two sides (of the equation) by some suitable quantities, other suitable quantities should be added to them so that the side containing the unknown becomes capable of yielding a root (''pada-prada''). The equation should then be formed again with the root of this side and the root of the known side. Thus the value of the unknown is obtained from that equation.
 This rule has been further elucidated by the author as follows: .
@@ Line 759: / Line 762: @@
 "When after perfect clearance of the two sides, there remain on one side the square, etc., of the unknown and on the other side the absolute term only, then, both the sides should be multiplied or divided by some suitable optional quantity; some equal quantities should further be added to or subtracted from both the sides so that the unknown side will become capable of yielding a root. The root of that side must be equal to the root of the absolute terms on the other side. For, by simultaneous equal additions, etc., to the two equal sides the equality remains. So forming an equation again with these roots the value of the unknown is found."
-It may be noted that in his treatise on arithmetic Bhāskara II has always followed the modern method of dividing by the coefficient of the square of the unknown.
+Bhāskara I in his treatise on arithmetic has always followed the modern method of dividing by the coefficient of the square of the unknown.
 Jnanaraja (1503) and Ganesa (1545) describe the same general methods for solving the quadratic as Bhāskara II.
-'''Elimination of the Middle Term.''' The method of solving the quadratic was known amongst the Hindu algebraists by the technical designation ''madhyamāharaṇa'' or "The Elimination of the Middle" (from ''madhyama'' = middle and ''aharana'' = removal, or destroying, that is, elimination). The origin of the name will be easily found in the principle underlying the method. By it a quadratic equation which, in its general form, contains three terms and so has a middle term, is reduced to a pure quadratic equation or a simple equation involving only two terms and so having no middle term. Thus the middle term of the original quadratic is eliminated by the method generally adopted for its solution. And hence the name. Bhāskara II has observed, "It is also specially designated by the learned teachers as the ''madhyamāharaṇa''. For by it, the removal of one of the two terms of the quadratic, the middle one, takes place." The name is, however, employed also in an extended sense so as to embrace the methods for solving the cubic and the biquadratic, where also certain terms are eliminated. It occurs as early as the works of Brahmagupta (628).
+'''Elimination of the Middle Term.''' ''madhyamāharaṇa'' or "The Elimination of the Middle" (from ''madhyama'' = middle and ''aharana'' =  elimination), the technical designation through which Hindu algebraists given the method to solve the quadratic equation.
+The name originated from the principle underlying the method.
+In general Quadratic equation contains three terms which has a middle term. By this method it will be converted to simple equations with only two terms ,where in middle term is eliminated. Hence the name ''madhyamāharaṇa''
+Bhāskara II has observed, "It is also specially designated by the learned teachers as the ''madhyamāharaṇa''. For by it, the removal of one of the two terms of the quadratic, the middle one, takes place." The name is, however, employed also in an extended sense so as to embrace the methods for solving the cubic and the biquadratic, where also certain terms are eliminated. It occurs as early as the works of Brahmagupta (628).
 '''Two Roots of the Quadratic'''. The Hindus recognized early that the quadratic has generally two root. In this connection Bhāskara II has quoted the following rule from an ancient writer of the name of Padmanabha whose treatise on algebra is not available now. "if after extracting roots the square-root of the absolute side of the quadratic be less than the negative absolute term on the other side, then taking it negative as well as positive, two values of the unknown are found."
@@ Line 786: / Line 795: @@
 == Equations Of Higher Degrees ==
-'''Cubic and Biquadratic.''' The Hindus did not achieve much in the solution of the cubic and biquadratic equations. Bhāskara II (1150) attempted the application of the method of the ''madhyamāharaṇa'' (elimination of the middle) to those equations also so as to reduce them by means of advantageous transformations and introduction of auxiliary quantities to simple and quadratic equations respectively. He thus anticipated one of the modern methods of solving the biquadratic. "If, however," observes Bhāskara II, "due to the presence of the cube, biquadrate, etc., the work of reduction cannot proceed any further, after the performance of such operations, for want of a root of the unknown side (of an equation), then the value of the unknown must be obtained by the ingenuity (of the mathematician)." He has given two examples, one of the cubic and the other of the biquadratic, in which such reduction is possible.
+'''Cubic and Biquadratic.'''
+There is no much achievement by Hindus in the solution of the cubic and biquadratic equations. Bhāskara II (1150) tried the application of the ''madhyamāharaṇa'' (elimination of the middle) method to those equations also so as to reduce them by means of advantageous transformations and introduction of auxiliary quantities to simple and quadratic equations respectively. He thus anticipated one of the modern methods of solving the biquadratic. "If, however," observes Bhāskara II, "due to the presence of the cube, biquadrate, etc., the work of reduction cannot proceed any further, after the performance of such operations, for want of a root of the unknown side (of an equation), then the value of the unknown must be obtained by the ingenuity (of the mathematician)." He has given two examples, one of the cubic and the other of the biquadratic, in which such reduction is possible.
 Example 1. "What is that number, which being multiplied by twelve and increased by the cube of the number, is equal to six times the square of the number added with thirty-five.
-Solution. "Here the number is x. This multiplied by twelve and increased by the cube of the number becomes x³ + 12x. It is equal to 6x² + 35. On making clearance, there appears on the first side x³ - 6x²+ 12x; on .the other side 35 . Adding negative eight to both the sides and extracting cube-roots, we get x - 2. = 0x + 3. And from this equation the number is found to be 5.
+Solution. "Here the number is x. This multiplied by twelve and increased by the cube of the number becomes x³ + 12x. It is equal to 6x² + 35. On making clearance, on one side  x³ - 6x²+ 12x; on the other side 35 . Adding negative eight to both the sides and extracting cube-roots, we get x - 2. = 0x + 3. And from this equation the number is 5.
-Example 2. "What is that number which being multiplied by 200 and added to the square of the number, and then multiplied by 2 and subtracted from the fourth power of the number will become one myriad less unity? Tell that number .
+Example 2. "What is that number which being multiplied by 200 and added to the square of the number, and then multiplied by 2 and subtracted from the fourth power of the number will become one myriad less unity?
-Solution. "Here the number is x; multiplied by 200 it becomes 200x; added to the square of the number, becomes x² + 200x; this being multiplied by two, 2x² + 400x; by this being diminished the fourth power of the number, namely, this x<sup>4</sup> becomes x<sup>4</sup>- 2x² - 400x. This is equal to a myriad less unity. Equiclearance having been made, the two sides will, be
+Solution. "Here the number is x; multiplied by 200 it becomes 200x; added to the square of the number, becomes x² + 200x; this being multiplied by two, 2x² + 400x; by this being diminished the fourth power of the number, namely, this x<sup>4</sup> becomes x<sup>4</sup>- 2x² - 400x. This is equal to a myriad less unity. Equi-clearance having been made, the two sides will, be
 x<sup>4</sup>- 2x² - 400x. = 0x<sup>4</sup> + 0x² + 0x + 9999.
-Here on adding four hundred x plus unity to the first side, the root can be extracted, but on adding the same to the other side, there will be no root of it. Thus the work (of reduction) does not proceed. Hence here , ingenuity (is called for). Here adding to both the sides four times the square of x, four hundred x and unity and then extracting roots, we get
+Here on adding four hundred x plus unity to the first side, the root can be extracted, but on adding the same to the other side, there will be no root of it. Thus the work (of reduction) does not proceed.  Here adding to both the sides four times the square of x, four hundred x and unity and then extracting roots, we get
 x² + 0x+ 1 = 0x² + 2x + 100.
-Again, forming equation with these and proceeding as before, the value of x is obtained as 11 In similar instances the value of the unknown must be determined
+Again, forming equation with these and proceeding as before, the value of x is obtained as 11."
-by the ingenuity of the mathematician."
 == Simultaneous Quadratic Equations ==

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