To solve the quadratic equation [tex]\(2x^2 - 8x = -7\)[/tex], let's follow these steps:
1. Rewrite the equation in standard form:
[tex]\[
2x^2 - 8x + 7 = 0
\][/tex]
2. Identify the coefficients:
The equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex] where:
[tex]\[
a = 2, \quad b = -8, \quad c = 7
\][/tex]
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-8)^2 - 4 \cdot 2 \cdot 7 = 64 - 56 = 8
\][/tex]
4. Find the square root of the discriminant:
[tex]\[
\sqrt{\Delta} = \sqrt{8} = 2\sqrt{2}
\][/tex]
5. Use the quadratic formula to find the solutions:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\sqrt{\Delta}\)[/tex]:
[tex]\[
x = \frac{8 \pm 2\sqrt{2}}{4}
\][/tex]
Simplifying further:
[tex]\[
x = \frac{8}{4} \pm \frac{2\sqrt{2}}{4} = 2 \pm \frac{\sqrt{2}}{2}
\][/tex]
Therefore, the solutions to the equation [tex]\(2x^2 - 8x + 7 = 0\)[/tex] are:
[tex]\[
x = 2 + \frac{\sqrt{2}}{2} \quad \text{and} \quad x = 2 - \frac{\sqrt{2}}{2}
\][/tex]
This matches the option [tex]\(\boxed{2 \pm \frac{\sqrt{2}}{2}}\)[/tex].
