To solve the quadratic equation [tex]\(x^2 + 7x = 8\)[/tex], we first need to bring it to the standard form of [tex]\(ax^2 + bx + c = 0\)[/tex].
Starting with the given equation:
[tex]\[ x^2 + 7x = 8 \][/tex]
Subtract 8 from both sides to get:
[tex]\[ x^2 + 7x - 8 = 0 \][/tex]
Now, we need to solve for the roots of this quadratic equation. These roots are found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In this equation, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are:
[tex]\[ a = 1, \quad b = 7, \quad c = -8 \][/tex]
Substitute these values into the quadratic formula:
[tex]\[ x = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-7 \pm \sqrt{49 + 32}}{2} \][/tex]
[tex]\[ x = \frac{-7 \pm \sqrt{81}}{2} \][/tex]
[tex]\[ x = \frac{-7 \pm 9}{2} \][/tex]
This gives us two solutions:
1. When we use the positive square root:
[tex]\[ x = \frac{-7 + 9}{2} \][/tex]
[tex]\[ x = \frac{2}{2} \][/tex]
[tex]\[ x = 1 \][/tex]
2. When we use the negative square root:
[tex]\[ x = \frac{-7 - 9}{2} \][/tex]
[tex]\[ x = \frac{-16}{2} \][/tex]
[tex]\[ x = -8 \][/tex]
Thus, the solutions to the equation are:
[tex]\[ x = 1 \quad \text{and} \quad x = -8 \][/tex]
Looking at the given options, we find that the pair of solutions matches option C.
Therefore, the correct answer is:
C. [tex]\( x = 1 ; x = -8 \)[/tex]
