Question

How do you solve ${x^2} + 8x = 24$ using the quadratic formula?

Answer 1

Hint:
According to given in the question we have to solve ${x^2} + 8x = 24$using the quadratic formula So, first of all to determine the solution or the roots/zeros of the quadratic expression we have to rearrange the terms of the quadratic expression which can be done by subtracting 24 in the both sides if the given expression.
Now, we have to determine the determine the values of a, b, and c which are the coefficient of \[{x^2}\], coefficient of x and the constant term which can be done by comparing the general form of the quadratic expression which is $a{x^2} + bx + c = 0$.
Now, to determine the roots of the quadratic expression we have to use the quadratic formula which is as mentioned below:

Formula used:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}..............(A)$
Where, a, b, and c are the coefficient of\[{x^2}\], coefficient of x and the constant terms.
Now, we have to solve the expression obtained after substituting all the values in the formula (A) to obtain both of the required roots/zeroes.

Complete step by step solution:
Step 1: First of all to determine the solution or the roots/zeros of the quadratic expression we have to rearrange the terms of the quadratic expression which can be done by subtracting 24 in both sides of the given expression. Hence,
$
   \Rightarrow {x^2} + 8x - 24 = 24 - 24 \\
   \Rightarrow {x^2} + 8x - 24 = 0 \\
 $
Step 2: Now, we have to determine the determine the values of a, b, and c which are the coefficient of \[{x^2}\], coefficient of x and the constant term which can be done by comparing the general form of the quadratic expression which is $a{x^2} + bx + c = 0$. Hence,
$ \Rightarrow a = 1,b = 8$and,
$ \Rightarrow c = - 24$
Step 3: Now, to determine the roots of the quadratic expression we have to use the quadratic formula (A) which is as mentioned in the solution hint. Hence,
$ \Rightarrow x = \dfrac{{ - 8 \pm \sqrt {{{(8)}^2} - 4 \times 1 \times ( - 24)} }}{{2 \times 1}}$
Step 4: Now, we have to solve the expression obtained after substituting all the values in the formula (A) to obtain both of the required roots/zeroes. Hence,
$
   \Rightarrow x = \dfrac{{ - 8 \pm \sqrt {64 + 96} }}{2} \\
   \Rightarrow x = \dfrac{{ - 8 \pm \sqrt {160} }}{2} \\
   \Rightarrow x = \dfrac{{ - 8 \pm 4\sqrt {10} }}{2} \\
 $
On eliminating the integers which can be eliminated in the expression as obtained just above,
$ \Rightarrow x = - 4 \pm 2\sqrt {10} $

Hence, with the help of the quadratic formula (A) we have determine the solution of the quadratic expression which is$x = - 4 \pm 2\sqrt {10} $

Note:
1) It is necessary that we have to obtain values of a, b, and c which are the coefficient of \[{x^2}\], coefficient of x and the constant term which can be done by comparing the general form of the quadratic expression which is $a{x^2} + bx + c = 0$.
2) On solving a quadratic expression only two possible roots/zeroes can be obtained which will satisfy the given quadratic expression mean on placing these in place of x the whole expression becomes 0.