To find the solutions to the quadratic equation
[tex]\[ 4x^2 - 10 = 10 - 20x, \][/tex]
we begin by putting all terms on one side to form a standard quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
First, let's rearrange the equation:
[tex]\[ 4x^2 - 10 = 10 - 20x. \][/tex]
Subtract [tex]\(10\)[/tex] from both sides:
[tex]\[ 4x^2 - 10 - 10 = -20x. \][/tex]
This simplifies to:
[tex]\[ 4x^2 - 20 = -20x. \][/tex]
Add [tex]\(20x\)[/tex] to both sides:
[tex]\[ 4x^2 + 20x - 20 = 0. \][/tex]
We now have a quadratic equation in the form:
[tex]\[ 4x^2 + 20x - 20 = 0. \][/tex]
To solve this quadratic equation, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \][/tex]
where [tex]\(a = 4\)[/tex], [tex]\(b = 20\)[/tex], and [tex]\(c = -20\)[/tex].
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-20 \pm \sqrt{20^2 - 4(4)(-20)}}{2(4)}. \][/tex]
First, calculate the discriminant:
[tex]\[ b^2 - 4ac = 20^2 - 4(4)(-20)
= 400 + 320
= 720. \][/tex]
So we have:
[tex]\[ x = \frac{-20 \pm \sqrt{720}}{8}. \][/tex]
Simplify the square root of [tex]\(720\)[/tex]:
[tex]\[ \sqrt{720} = \sqrt{144 \times 5} = 12\sqrt{5}. \][/tex]
Substitute back into the quadratic formula:
[tex]\[ x = \frac{-20 \pm 12\sqrt{5}}{8}. \][/tex]
Simplify the fraction:
[tex]\[ x = \frac{-20}{8} \pm \frac{12\sqrt{5}}{8}
= -\frac{20}{8} \pm \frac{12\sqrt{5}}{8}
= -\frac{5}{2} \pm \frac{3\sqrt{5}}{2}. \][/tex]
Therefore, the solutions to the quadratic equation are:
[tex]\[ x = -\frac{5}{2} + \frac{3\sqrt{5}}{2} \quad \text{and} \quad x = -\frac{5}{2} - \frac{3\sqrt{5}}{2}. \][/tex]
The correct answer is:
[tex]\[ \boxed{D. \; x = \frac{-5 \pm 3\sqrt{5}}{2}} \][/tex]
