Question

Use the quadratic formula to solve the quadratic equation 10x2+5x=5x−8 . Express its solutions in the form a±bi .(1 point) Responses 0±2√5i 0 plus or minus the fraction with numerator square root of 2 and denominator 5 i 0±25√5i 0 plus or minus the fraction with numerator 2 square root of 5 and denominator 5 i 0±5√5i 0 plus or minus the fraction with numerator square root of 5 and denominator 5 i 0±25i

Answer 1

To solve the quadratic equation \( 10x^2 + 5x = 5x - 8 \), we first rearrange it into the standard form:

\[
10x^2 + 5x - 5x + 8 = 0 \implies 10x^2 + 8 = 0
\]

Simplifying this gives us:

\[
10x^2 + 8 = 0 \implies 10x^2 = -8 \implies x^2 = -\frac{8}{10} = -\frac{4}{5}
\]

This indicates that \( x^2 \) is negative, so the solutions will involve imaginary numbers.

Next, we apply the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In our case, \( a = 10 \), \( b = 0 \), and \( c = 8 \).

Calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = 0^2 - 4(10)(8) = -320
\]

Now substitute into the quadratic formula:

\[
x = \frac{-0 \pm \sqrt{-320}}{2(10)} = \frac{\pm \sqrt{-320}}{20}
\]

Since \( \sqrt{-320} \) can be expressed as \( \sqrt{320}i \), we proceed to simplify \( \sqrt{320} \):

\[
\sqrt{320} = \sqrt{64 \cdot 5} = \sqrt{64} \cdot \sqrt{5} = 8\sqrt{5}
\]

Now substituting this back in:

\[
x = \frac{\pm 8\sqrt{5} i}{20}
\]

Simplifying further:

\[
x = \frac{\pm 2\sqrt{5} i}{5}
\]

Thus, the solutions to the equation \( 10x^2 + 5x = 5x - 8 \) are:

\[
x = 0 \pm \frac{2\sqrt{5}}{5} i
\]

Therefore, the final answer is:

\[
0 \pm \frac{2\sqrt{5}}{5} i
\]