What are the solutions to [tex]\( x^2 + 8x + 7 = 0 \)[/tex]?

A. [tex]\( x = -8 \)[/tex] and [tex]\( x = -7 \)[/tex]

B. [tex]\( x = -7 \)[/tex] and [tex]\( x = -1 \)[/tex]

C. [tex]\( x = 1 \)[/tex] and [tex]\( x = 7 \)[/tex]

D. [tex]\( x = 7 \)[/tex] and [tex]\( x = 8 \)[/tex]



Answer :

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]