To find the solutions to the quadratic equation [tex]\(x^2 + 8x + 7 = 0\)[/tex], we can use the quadratic formula, which is given by:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Here, the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are [tex]\(a = 1\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = 7\)[/tex], respectively.
1. Calculate the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is calculated as:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 8^2 - 4(1)(7) \][/tex]
[tex]\[ \Delta = 64 - 28 \][/tex]
[tex]\[ \Delta = 36 \][/tex]
2. Find the Square Root of the Discriminant:
The square root of the discriminant is:
[tex]\[ \sqrt{\Delta} = \sqrt{36} = 6 \][/tex]
3. Calculate the Two Solutions:
Using the quadratic formula, we find the two solutions:
[tex]\[ x_{1, 2} = \frac{{-b \pm \sqrt{\Delta}}}{2a} \][/tex]
Substitute [tex]\(b = 8\)[/tex], [tex]\(\sqrt{\Delta} = 6\)[/tex], and [tex]\(a = 1\)[/tex]:
[tex]\[
x_1 = \frac{{-8 + 6}}{2(1)} = \frac{{-8 + 6}}{2} = \frac{{-2}}{2} = -1
\][/tex]
[tex]\[
x_2 = \frac{{-8 - 6}}{2(1)} = \frac{{-8 - 6}}{2} = \frac{{-14}}{2} = -7
\][/tex]
Thus, the solutions to the quadratic equation [tex]\(x^2 + 8x + 7 = 0\)[/tex] are:
[tex]\[ x = -1 \quad \text{and} \quad x = -7 \][/tex]
Therefore, the correct answer is:
[tex]\[ x = -7 \quad \text{and} \quad x = -1 \][/tex]
