Certainly! Let's solve the equation [tex]\( x^2 - 14x = -40 \)[/tex] by completing the square step-by-step.
1. Start with the given equation:
[tex]\[
x^2 - 14x = -40
\][/tex]
2. First, we'll move the [tex]\( -40 \)[/tex] to the left side to set the equation to zero:
[tex]\[
x^2 - 14x + 40 = 0
\][/tex]
3. Now, let's complete the square on [tex]\( x^2 - 14x \)[/tex]. To do this, take half of the coefficient of [tex]\( x \)[/tex] (which is [tex]\(-14\)[/tex]), square it, and add and subtract this value inside the equation.
Half of [tex]\(-14\)[/tex] is:
[tex]\[
\frac{-14}{2} = -7
\][/tex]
Squaring [tex]\(-7\)[/tex] gives us:
[tex]\[
(-7)^2 = 49
\][/tex]
4. Add and subtract 49 inside the equation:
[tex]\[
x^2 - 14x + 49 - 49 + 40 = 0
\][/tex]
This can be written as:
[tex]\[
(x - 7)^2 - 9 = 0
\][/tex]
5. Isolate the perfect square term:
[tex]\[
(x - 7)^2 = 9
\][/tex]
6. Take the square root of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x - 7 = \pm 3
\][/tex]
This gives us two solutions:
[tex]\[
x - 7 = 3 \quad \text{or} \quad x - 7 = -3
\][/tex]
7. Solve for [tex]\(x\)[/tex] in each case:
[tex]\[
x - 7 = 3 \implies x = 10
\][/tex]
[tex]\[
x - 7 = -3 \implies x = 4
\][/tex]
So, the solutions to the equation [tex]\( x^2 - 14x = -40 \)[/tex] are [tex]\( x = 4 \)[/tex] and [tex]\( x = 10 \)[/tex].
Therefore, the correct answers are:
- 4
- 10
