To find the solutions to the quadratic equation [tex]\(4x^2 - 10 = 10 - 20x\)[/tex], follow these steps:
1. Move all terms to one side to set the equation to 0:
[tex]\[
4x^2 - 10 - 10 + 20x = 0 \implies 4x^2 + 20x - 20 = 0
\][/tex]
2. Identify the coefficients for the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
a = 4, \quad b = 20, \quad c = -20
\][/tex]
3. Calculate the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[
\Delta = b^2 - 4ac = 20^2 - 4 \cdot 4 \cdot (-20) = 400 + 320 = 720
\][/tex]
4. Use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{\Delta}}{2a}\)[/tex] to find the solutions:
[tex]\[
x = \frac{-20 \pm \sqrt{720}}{8}
\][/tex]
5. Simplify the expression:
[tex]\[
\sqrt{720} = 12 \sqrt{5}
\][/tex]
Substituting back, we get:
[tex]\[
x = \frac{-20 \pm 12\sqrt{5}}{8} = \frac{-20}{8} \pm \frac{12 \sqrt{5}}{8}
\][/tex]
Simplifying further, we get:
[tex]\[
x = -2.5 \pm 1.5\sqrt{5}
\][/tex]
6. Convert the expressions to match the provided choices:
[tex]\[
x = -\frac{20}{8} \pm \frac{12\sqrt{5}}{8} \implies x = -\frac{5}{2} \pm \frac{3\sqrt{5}}{2}
\][/tex]
Hence, the solutions are:
[tex]\[
x = \frac{-5 \pm 3 \sqrt{5}}{2}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{D. \; x = \frac{-5 \pm 3 \sqrt{5}}{2}}
\][/tex]
