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This unit facilitates you in,

defining quadratic equation, pure and adfected quadratic equations.

solving pure quadratic equations.

solving quadratic equations by factorisation method.

solving quadratic equations by completing the square method.

deriving the formula to find the roots of quadratic equation.

using formula to solve quadratic equations.

drawing graphs for quadratic expressions.

solving quadratic equations graphically.

establising relation between roots and coefficients of quadratic equation.

framing quadratic equations.

finding the discriminant and interpret the nature of roots of quadratic equation.

Quadratic Equations

Quadratic expression and quadratic equation Pure and adfected quadratic equations Solution of quadratic equation by * Factorisation method * Completing the square method * Formula method * Graphical method Relation between roots and coefficients Forming quadratic equations

9

Brahmagupta (A.D 598-665, India)

Solving of quadratic equations in general form is often credited to ancient Indian mathe- maticians. Brahmagupta gave an expl icit formula to solve a quadratic equation of the form ax2 + bx = c. Later Sridharacharya (A.D. 1025) derived a formula, now known as the quadratic formula, for solving a quadratic equation by the method of completing the square.

An equation means nothing to me, unless it expresses a thought of God.

- Srinivas Ramanujan

194 UNIT-9

We are familiar with playing number games. Let us consider two such examples.

Example 1 Example 2 * Take a non-zero whole number. * Take a non-zero whole number

* Add 7 to it. * Add 7 to it.

* Equate it to 12. * Multiply the sum by the same whole number.

* What is the number? * Equate it to 12.

* What is the number?

How to find the numbers? Let the non-zero whole number be 'x'

If we follow the steps, we get

in example 1, x x + 7 and x + 7 = 12 ........(i)

in example 2, x x + 7 andx(x + 7) = 12 ........(ii)

Take equation (i), x + 7 = 12. Here, x is the variable, whose degree is 1.

It is a linear equation. It has only one root.

i.e., x + 7 = 12, x = 12 – 7, x = 5

Take equation (ii), x (x + 7) = 12, x2 + 7x = 12

Here, x is the variable, whose degree is 2. It is a quadratic equation. How to solve it? How many roots does a quadratic equation have?

It is very essential to learn this, because quadratic equations have wide applications in other branches of mathematics, in other subjects and also in real life situations.

For instance, suppose an old age home trust decides to build a prayer hall having floor area of 300 sq.m., with its length one meter more than twice its breadth . What should be the length and breadth of the hall?

Let, the breadth be x m. Then, its length will be (2x + 1)m

Its area = x(2x + 1)sq.m x(2x + 1) = 300 ( Given)

This information can be diagrammatically represented as follows.

We have, Area = x(2x + 1) = (2x2 + x)m2

So, 2x2 + x = 300. This is a quadratic equation.

Below are given some more illustrations in verbal statement form which when converted into equation form result in quadratic equation form.

Study the statements. Try to express each statement in equation form.

1. An express train takes one hour less than the passenger train to travel 132 km between Bangalore and Mysore. If the average speed of the express train is 11 km/hr more than that of the passenger train, what is the average speed of the two trains?

22 1 300 m mx x x

(2x + 1) m

Quadratic Equations 195

2. A cottage industry produces a certain number of wooden toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was ̀ 750. What is the number of toys produced on that day?

3. A motor boat whose speed is 18 km/hr in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. What is the speed of the stream?

4. Two water taps together can fill a tank in 9 3 8 hours. The tap of larger diameter

takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

How to solve these problems?

Know this!

It is believed that Babylonians were the first to solve quadratic equations. Greek mathematician Euclid developed a geometrical approach for finding lengths, which are nothing but solutions of quadratic equations.

Solving of quadratic equations, in general form, is often credited to ancient Indian mathematicians like Brahmagupta (A.D. 598-665) and Sridharacharya (A.D. 1025). An Arab mathematician Al-khwarizni (about A.D. 800) also studied quadratic equations of different types.

Abraham bar Hiyya Ha-Nasi, in his book "Liber embardorum" published in Europe in A.D. 1145 gave complete solutions of different quadratic equations.

In this unit, let us study quadratic equations, various methods of finding their roots and also applications of quadratic equations.

Quadratic equation

Recall that we have studied about quadratic polynomials in unit 8.

A polynomial of the form ax2 + bx + c, where a 0 is a quadratic polynomial or expression in the variable x of degree 2. If a quadratic expression ax2 + bx + c is equated to zero, it becomes a quadratic equation.

Below are given some verbal statements, when converted to quadratic expression form and further equated to zero become quadratic equations. Study the examples.

196 UNIT-9

Know this! The word quadratic is derived from the Latin word "quadratum" which means "A square figure".

So it can be stated that, if p(x) is a quadratic polynomial, then p(x) = 0 is a quadratic equation.

In fact any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2 is a quadratic equation whose standard form is ax2 + bx + c = 0, a 0.

A quadratic equation in the variable x is an equation of the form ax2 + bx + c = 0 where a, b, c, are real numbers and a 0. Characteristics of a quadratic equation are,

• it is an equation in one variable.

• it is an equation whose single variable is of degree 2.

• the standard form of quadratic equation is ax2 + bx + c = 0.

Here, a is the coefficient of x2,

b is the coefficient of x,

c is the constant term,

a, b, c are real numbers and a 0.

• the terms are written in descending order of the power of the variable.

In a quadratic equation why is a 0? What happens to the quadratic equation, if a = 0? Discuss in the class.

We have discussed that in a quadratic equation, a 0.

What happens to standard form of quadratic equation when b or c or both b and c are equal to zero?

Sl. Verbal statement Quadratic Quadratic No. expression equation

1. The sum of a number and five 5x2 + x 5x2 + x =0 times its squares.

2. A wire is bent to form the legs of a right angled triangle. If one of them is 2cm more than the other, what will be its area?

3. The runs scored by a cricket x2 – x – 6 x2 – x – 6 =0 team are 6 less than the difference of the runs scored by the first batsman and the square of runs scored by him

1 ( 2) 2

x x 21 ( 2 ) 0 2

x x

21 ( 2 ) 2

x x 2 2 0x x

Quadratic Equations 197

Observe the table given below.

Sl. Value of b value of c Result

1. b = 0 c 0 ax2 + c = 0

2. b 0 c = 0 ax2 + bx = 0

3 b = 0 c = 0 ax2 = 0

4. b 0 c 0 ax2 + bx + c = 0

Observe that, in all the above cases the equation remains as a quadratic equation.

ILLUSTRATIVE EXAMPLES

Example 1: Check whether the following are quadratic equations: (i) 2x + x2 + 1 = 0 (ii) 6x3 + x2 = 2

(iii) 3

( 8) 10 0 4

x x x (iv) x(x + 1) + 8 = (x + 2) (x – 2)

Sol. (i) 2x + x2 + 1 = 0 Arrange the terms in descending order of their powers. x2 + 2x + 1 = 0 It is in the standard form ax2 + bx +c = 0. The given equation is a quadratic equation.

(ii) 6x3 + x2 = 2 By rearranging the terms, we get 6x3 + x2 – 2 = 0 The highest degree of the variable is 3. The given equation is not a quadratic equation.

(iii) 3 ( 8) 10 0 4

x x x

By simplifying we get

2 23 248 10 0 4 4

x x x x

4x2 – 32x + 3x2 – 24x + 40 = 0 x2 – 56x + 40 = 0. It is of the form ax2 + bx + c = 0. it is a quadratic equation.

(iv) x (x + 1) + 8 = (x + 2)(x – 2) By simplifying we get

x2 + x + 8 = x2 – 4 2x 2x + x + 8 + 4 = 0, x + 12 = 0

It is not of the form ax2 + bx + c = 0. The variable x is only in the first degree. it is not a quadratic equation.

198 UNIT-9

EXERCISE 9.1

1. Check whether the following are quadratic equations:

(i) x2 – x = 0 (ii) x2 = 8 (iii) 2 1 0 2

x x (iv) 3x – 10 = 0

(v) 2 29 5 0 4

x x (vi) 2 25 6 5

x x (vii) 22 3 0x x (viii) 223 13

x

(ix) x3 – 10x + 74 = 0 (x) x2 – y 2 = 0 2. Simplify the following equations and check whether they are quadratic equations.

(i) x(x + 6) = 0 (ii) (x - 4)(2x – 3) = 0

(iii) (x + 9)(x – 9) = 0 (iv) (x + 2)(x – 7) = 5

(v) 3x + (2x – 1)(x – 9) = 0 (vi) (x + 1)2 = 2(x – 3)

(vii) (2x – 1)(x – 3) = (x +