Answer
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Hint: We will first split the middle as to get -24 as the product of those two split terms. Then we will get factors and equate them to 0 to get the desired answer.
Complete step-by-step answer:
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.
So, the correct answer is “Option D”.
Note: The student might make the mistake of finding a factor without considering the negative sign and will end up not getting any common factor among it.
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 $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.
Complete step-by-step answer:
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.
So, the correct answer is “Option D”.
Note: The student might make the mistake of finding a factor without considering the negative sign and will end up not getting any common factor among it.
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 $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.