Questions & Answers

Question

Answers

A. 3, 8

B. -3, -8

C. -3, 8

D. -8, 3

Answer
Verified

We will first know how to use the method of factorization.

Consider that we have an equation $a{x^2} + bx + c = 0$ ……….(1)

Then multiply a and c, we will get: ac

Now, we will find factors of ac such that the linear sum of those factors results in b.

Now, let us try to apply this method on ${x^2} + 5x - 24 = 0$.

Comparing this equation with equation (1), we will see that:-

a = 1, b = 5 and c = -24.

$a \times c = 1 \times ( - 24) = - 24$

Now, let us find all the factors of -24.

Factors = -1 and 24, -2 and 12, -3 and 8, -4 and 6, -6 and 4, -8 and 3, -12 and 2; and -24 and 1.

We see that -3 + 8 = 5.

We can rewrite our polynomial ${x^2} + 5x - 24 = 0$ as ${x^2} - 3x + 8x - 24 = 0$.

Now, taking $x$ common from first two terms and 8 common from the next two terms, we will have:-

$x(x - 3) + 8(x - 3) = 0$

Now, taking $x - 3$ common, we will have:-

$(x - 3)(x + 8) = 0$

We know that if a.b = 0, then either a = 0 or b = 0.

So, either $x - 3 = 0$ or $x + 8 = 0$.

Taking the constants on the right hand side in both equations:-

So, either $x = 3$ or $x = - 8$.

Hence, the roots are 3 and -8.

There is one more way of doing the same question by using direct formula which says the roots of the equation $a{x^2} + bx + c = 0$ are given by $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.