To solve the quadratic equation [tex]\( 4x^2 - 10 = 10 - 20x \)[/tex], let's follow these steps:
1. Rearrange the equation into standard form
The given quadratic equation is:
[tex]\[
4x^2 - 10 = 10 - 20x
\][/tex]
First, let's move all terms to one side to arrange it in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
4x^2 - 10 - 10 + 20x = 0
\][/tex]
Simplifying this, we get:
[tex]\[
4x^2 + 20x - 20 = 0
\][/tex]
2. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]
In the equation [tex]\( 4x^2 + 20x - 20 = 0 \)[/tex], the coefficients are:
[tex]\[
a = 4, \quad b = 20, \quad c = -20
\][/tex]
3. Compute the discriminant
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the coefficients, we get:
[tex]\[
\Delta = 20^2 - 4 \cdot 4 \cdot (-20)
\][/tex]
[tex]\[
\Delta = 400 + 320 = 720
\][/tex]
4. Solve using the quadratic formula
The solutions of the quadratic equation are given by:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values [tex]\(a = 4\)[/tex], [tex]\(b = 20\)[/tex], and [tex]\(\Delta = 720\)[/tex]:
[tex]\[
x = \frac{-20 \pm \sqrt{720}}{8}
\][/tex]
Simplify the square root:
[tex]\[
\sqrt{720} = \sqrt{144 \times 5} = 12\sqrt{5}
\][/tex]
Therefore, the solutions become:
[tex]\[
x = \frac{-20 \pm 12\sqrt{5}}{8}
\][/tex]
Simplifying further:
[tex]\[
x = \frac{-20}{8} \pm \frac{12\sqrt{5}}{8}
\][/tex]
[tex]\[
x = -\frac{5}{2} \pm \frac{3\sqrt{5}}{2}
\][/tex]
5. Interpret the solutions
The solutions in a more readable form are:
[tex]\[
x = \frac{-5 \pm 3\sqrt{5}}{2}
\][/tex]
Therefore, the correct answer is:
A. [tex]\( x = \frac{-5 \pm 3\sqrt{5}}{2} \)[/tex]
