The first few steps in solving the quadratic equation [tex]8x^2 + 80x = -5[/tex] by completing the square are shown:

[tex]\[
\begin{array}{l}
8x^2 + 80x = -5 \\
8(x^2 + 10x) = -5 \\
8(x^2 + 10x + 25) = -5 + \, \text{[missing number]}
\end{array}
\][/tex]

Which number is missing in the last step?

A. -200
B. -25
C. 25
D. 200



Answer :

Let's solve the equation step-by-step to determine the missing number.

1. Start with the given equation:
[tex]\[ 8x^2 + 80x = -5 \][/tex]

2. Factor out the 8 from the left-hand side:
[tex]\[ 8(x^2 + 10x) = -5 \][/tex]

3. To complete the square inside the parentheses, we need to add and subtract a certain number. The goal is to turn [tex]\(x^2 + 10x\)[/tex] into a perfect square trinomial.
[tex]\[ x^2 + 10x + (constant) \][/tex]

4. The constant can be found by taking half of the coefficient of [tex]\(x\)[/tex], which is 10, and then squaring it:
[tex]\[ \left(\frac{10}{2}\right)^2 = 5^2 = 25 \][/tex]

5. Add this constant inside the parentheses:
[tex]\[ 8(x^2 + 10x + 25) \][/tex]

6. However, we can't just add 25 inside the parentheses without making an equivalent adjustment on the right-hand side. Since we added [tex]\(25\)[/tex] inside the parentheses, which is being multiplied by 8, we need to add [tex]\(8 \times 25\)[/tex] to the right-hand side of the equation as well:
[tex]\[ 8(x^2 + 10x + 25) = -5 + 8 \times 25 \][/tex]

7. Calculate [tex]\(8 \times 25\)[/tex]:
[tex]\[ 8 \times 25 = 200 \][/tex]

Thus, the missing number is:
[tex]\[ 200 \][/tex]

So the missing number in the last step is [tex]\(200\)[/tex].