Question

The value of m for which the difference between the roots of equation x2+mx+8=0 is 2 are

Answer 1

Answer:

Solution;

Note:

If we consider a quadratic equation in variable x , say: ax^2 + bx + c = 0,

Let A and B be its roots, then:

A+B = - b/a

A•B = c/a

Here, the given quadratic equation is;

x^2 + mx + 8 = 0

Clearly,

Here we have;

a = 1

b = m

c = 8

Let us assume that , A and B are the roots of the given quadratic equation,

Thus;

A+B = - b/a = - m/1 = - m

A•B = c/a = 8/1 = 8

Also, it is given that ;

|A-B| = 2

ie, A-B = ±2

Note:

(a + b)^2 = (a - b)^2 + 4•a•b

We can use this identity to find the value of m.

Now;

=> (A + B)^2 = (A - B)^2 + 4•A•B

=> (- m)^2 = (± 2)^2 + 4•8

=> m^2 = 4 + 32

=> m^2 = 36

=> m = ± √36

=> m = ± 6

Moreover ;

There will be two different equations corresponding to two different values of m.

Case:(1)

When , m = 6 ;

Then , the quadratic equation will be ;

x^2 + 6x + 8 = 0

Case:(2)

When , m = - 6

Then , the quadratic equation will be ;

x^2 - 6x + 8 = 0

Verification :

Case:(1)

=> x^2 + 6x + 8 = 0

=> x^2 + 4x + 2x + 8 = 0

=> x(x + 4) + 2(x + 4) = 0

=> (x + 4)(x + 2) = 0

=> x = - 4 , - 2

Here, the difference between the roots

= - 2 - (- 4)

= - 2 + 4

= 2

Case:(2)

=> x^2 - 6x + 8 = 0

=> x^2 - 4x - 2x + 8 = 0

=> x(x - 4) + 2(x - 4) = 0

=> (x - 4)(x - 2) = 0

=> x = 4 , 2

Here, the difference between the roots

= 4 - 2

= 2

In Both the case, the difference between the roots is 2 , Hence verified.