To solve the quadratic equation [tex]\( x^2 + 20x + 100 = 36 \)[/tex], let's follow these steps:
### Step 1: Simplify the equation
First, we need to move all terms to one side of the equation to set it equal to zero.
[tex]\[ x^2 + 20x + 100 - 36 = 0 \][/tex]
Simplify the constants on the left-hand side:
[tex]\[ x^2 + 20x + 64 = 0 \][/tex]
### Step 2: Identify coefficients
The simplified equation is in the standard form of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], where:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 20\)[/tex]
- [tex]\(c = 64\)[/tex]
### Step 3: Use the quadratic formula
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[ x = \frac{-20 \pm \sqrt{20^2 - 4 \cdot 1 \cdot 64}}{2 \cdot 1} \][/tex]
### Step 4: Calculate the discriminant
Calculate the value inside the square root (the discriminant):
[tex]\[ b^2 - 4ac = 20^2 - 4 \cdot 1 \cdot 64 \][/tex]
[tex]\[ = 400 - 256 \][/tex]
[tex]\[ = 144 \][/tex]
### Step 5: Solve for [tex]\(x\)[/tex]
Now plug the discriminant back into the quadratic formula:
[tex]\[ x = \frac{-20 \pm \sqrt{144}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-20 \pm 12}{2} \][/tex]
### Step 6: Compute the two possible solutions
Solve for the two values of [tex]\(x\)[/tex]:
[tex]\[ x_1 = \frac{-20 + 12}{2} = \frac{-8}{2} = -4 \][/tex]
[tex]\[ x_2 = \frac{-20 - 12}{2} = \frac{-32}{2} = -16 \][/tex]
Thus, the solutions to the equation are [tex]\(x = -4\)[/tex] and [tex]\(x = -16\)[/tex].
### Step 7: Identify the correct multiple-choice answer
From the given multiple-choice options:
- [tex]\(x = -16\)[/tex] or [tex]\(x = -4\)[/tex]
- [tex]\(x = -10\)[/tex]
- [tex]\(x = -8\)[/tex]
- [tex]\(x = 4\)[/tex] or [tex]\(x = 16\)[/tex]
The correct choice corresponds to the option where [tex]\(x = -16\)[/tex] or [tex]\(x = -4\)[/tex].
Therefore, the answer is:
[tex]\[
\boxed{1}
\][/tex]
