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