To solve the equation [tex]\( x^2 + 14x = -24 \)[/tex] by completing the square, follow these steps:
1. Rewrite the Equation:
Start with the equation given:
[tex]\[
x^2 + 14x = -24
\][/tex]
2. Move the Constant to the Right Side:
The constant term [tex]\(-24\)[/tex] is already on the right side, so this step is already done.
3. Complete the Square:
To complete the square, find the term that needs to be added to both sides of the equation to form a perfect square trinomial on the left side.
- Take half of the coefficient of [tex]\(x\)[/tex], which is 14, divide it by 2 to get 7, and then square it:
[tex]\[
\left(\frac{14}{2}\right)^2 = 7^2 = 49
\][/tex]
- Add 49 to both sides of the equation:
[tex]\[
x^2 + 14x + 49 = -24 + 49
\][/tex]
- Simplify the right side:
[tex]\[
x^2 + 14x + 49 = 25
\][/tex]
4. Write the Left Side as a Perfect Square:
Now the left side can be written as a binomial squared:
[tex]\[
(x + 7)^2 = 25
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], take the square root of both sides of the equation:
[tex]\[
x + 7 = \pm 5
\][/tex]
This gives us two equations to solve:
[tex]\[
x + 7 = 5 \quad \text{or} \quad x + 7 = -5
\][/tex]
6. Solve Each Equation:
- For [tex]\( x + 7 = 5 \)[/tex]:
[tex]\[
x = 5 - 7 = -2
\][/tex]
- For [tex]\( x + 7 = -5 \)[/tex]:
[tex]\[
x = -5 - 7 = -12
\][/tex]
So the solutions are [tex]\( -2 \)[/tex] and [tex]\( -12 \)[/tex]. Therefore, the solution set of the equation [tex]\( x^2 + 14x = -24 \)[/tex] is:
[tex]\[
\{-2, -12\}
\][/tex]
