Quadratic Equations Complete Tutorial - 1 - CAT 2009 Online Preparation

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CAT 2009 Online Preparation

Quadratic Equations Complete Tutorial - 1

Author: Sureshbala

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QUADRATIC EQUATIONS

"If a variable occurs in an equation with one positive integer powers and the highest power is two, then it is called a Quadratic Equation (in that variable)."

In other words, a second degree polynomial in x equated to zero will be a quadratic equation. For such an equation to be a quadratic equation, the co-efficient x^2 should not be zero.

The most general form of a quadratic equation is ax^2+bx+c=0 , where $a \not =0$ (and a, b, c are real).

.....(1)

.....(2)

.....(4)

Like a first degree equation in x has one value of x satisfying the equation, a quadratic equation in x will have TWO values of x that satisfy the equation. The values of x that. Satisfy the equation are called the ROOTS of the equation. These roots may be real or imaginary.

For the four quadratic equations given above, the roots are as given below.

Equation (1): x = 2 and x = 3

Equation (2): x = -2 and x = 3

Equation (3): x =1/2 and x= -2

Equation (4): x = 1 and x = -3/2

In general, the roots of a quadratic equation can be found out in two ways

(i) by factorizing the expression on the left hand side of the quadratic equation

(ii) by using the standard formula

All the expressions may not be easy to factorise whereas applying the formula is simple and straight forward.

Finding the roots by factorization

If the quadratic equation ax^2+bx+c=0 can be written in the form $(x- \alpha )(x- \beta)=0,$ then the roots of equation are $\alpha \, and \, \beta$

To find the roots of a quadratic equation, we should first write it in the form $(x- \alpha )(x- \beta)=0,$ i.e., the left hand side ax^2+bx+c=0 should be factorised in two factors.

For this purpose, we should go through the following steps. We will understand these steps with the help of the equation x^2-5x+6=0 which is the first of the four quadratic equations we looked at as examples above.

- First write down b (the co-efficient of x) as the sum of two quantities whose product is equal to ac. In this case -5 has to be written as the sum of two quantities whose product is 6. We can write -5 as (-3) + (-2) so that the product of (-3) and (-2) is equal to 6.

- Now rewrite the equation with the 'bx' term split in the above manner. In this case, the given equation can be written as x^2-3x-2x+6=0

- Take the first two terms and rewrite them together after taking out the common factor between the two of them. Similarly, the third and fourth terms should be rewritten after taking out the common factor between the two of them. In the process, you should ensure that what is left from the first and second terms (after removing the common factor) is the same as that left from the third and fourth term (after removing their common factor).

In this case, the equation can be rewritten as x(x-3)-2(x-3)=0; between the first and second terms as well as the third and fourth terms, we are left with (x-3) is a common factor.

- Rewrite the entire left hand side to get the form $(x- \alpha)(x- \beta)$

In this case, if we take out (x - 3) as the common factor, we can rewrite the given equation as (x - 3)(x - 2) = 0

- Now $\alpha \, and \, \beta$ are the roots of the quadratic equation.

In the example we have taken, the roots of the equation are 3 and 2.

For the other three quadratic equations given above as examples, let us see how to factorise the expression and the roots.

For equation (2), i.e. , x^2-x-6=0 the co-efficient of x which is -1 can be rewritten as (-3) + (-2) so that their product is -6 which is equal to ac (1 multiplied by -6). Then we can rewrite the equation as (x - 3) (x + 2) = 0 giving us the roots as 3 and -2.

For equation (3), i.e. , 2x^2+3x-2=0, , the co-efficient of x which is 3 can be re written as (+4) + (-1) so that their product is -4 which is the value of ac (-2 multiplied by 2). Then we can rewrite the equation as (2x - 1) (x + 2) =0 giving the roots as 1/2 and -2.

For equation (4), i.e. , 2x^2+x-3=0 , the co-efficient of x which is 1 can be rewritten as (+3) + (-2) so that their product is -6 which is equal to ac (2 multiplied by -3). Then we can rewrite the given equation as (x - 1)(2x + 3) = 0 giving us the roots as 1 and -3/2.

Finding the roots by using the formula

If the quadratic equation is ax^2+bx+c=0 , then we can use the standard formula given below to find out the roots of the equation.

$x= \cfrac {-b \pm \sqrt{b^2-4ac}}{2a}$

The roots of the four quadratic equations we took as examples above can be taken and their roots found out by using the above formula. The student is advised to check it out for himself that the roots can be obtained by using this formula also.

SUM AND PRODUCT OF ROOTS OF A QUADRATIC EQUATION

For the quadratic equation ax^2+bx+c-0 , the sum of the roots and the product of the roots can be given by the following.