To determine the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( x^2 + 2x = 24 \)[/tex], follow these steps:
1. Rewrite the equation: Start with the given equation and bring all terms to one side to set the equation to zero.
[tex]\[
x^2 + 2x = 24
\][/tex]
Subtract 24 from both sides to get:
[tex]\[
x^2 + 2x - 24 = 0
\][/tex]
2. Factor the quadratic equation: We need to factor the equation [tex]\( x^2 + 2x - 24 = 0 \)[/tex]. Look for two numbers that multiply to [tex]\(-24\)[/tex] and add to [tex]\(2\)[/tex]. These numbers are [tex]\(6\)[/tex] and [tex]\(-4\)[/tex].
[tex]\[
(x + 6)(x - 4) = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex] by setting each factor to zero:
[tex]\[
x + 6 = 0 \quad \text{or} \quad x - 4 = 0
\][/tex]
Solving these equations gives:
[tex]\[
x = -6 \quad \text{or} \quad x = 4
\][/tex]
Therefore, the solutions to the equation [tex]\( x^2 + 2x = 24 \)[/tex] are [tex]\( x = -6 \)[/tex] and [tex]\( x = 4 \)[/tex].
Thus, the correct answer is:
[tex]\[
4 \text{ and } -6
\][/tex]
