To solve the quadratic equation \(10x^2 + 5x = 5x - 8\), we first need to rearrange it into standard form \(ax^2 + bx + c = 0\).
Starting with the original equation:
\[ 10x^2 + 5x - 5x + 8 = 0 \]
This simplifies to:
\[ 10x^2 + 8 = 0 \]
Now we can rewrite the equation in standard form:
\[ 10x^2 + 0x + 8 = 0 \]
Here, \(a = 10\), \(b = 0\), and \(c = 8\).
Next, we will use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \(a\), \(b\), and \(c\) into the formula:
- Calculate \(b^2 - 4ac\):
\[ b^2 - 4ac = 0^2 - 4(10)(8) = 0 - 320 = -320 \]
- Now, substitute into the quadratic formula:
\[ x = \frac{-0 \pm \sqrt{-320}}{2(10)} \]
This simplifies to:
\[ x = \frac{\pm \sqrt{-320}}{20} \]
- Since \(\sqrt{-320} = \sqrt{320}i = \sqrt{64 \cdot 5}i = 8\sqrt{5}i\), we can express the equation as:
\[ x = \frac{\pm 8\sqrt{5}i}{20} \]
This simplifies to:
\[ x = \frac{\pm 2\sqrt{5}i}{5} \]
Thus, we can express the solutions as:
\[ x = 0 \pm \frac{2\sqrt{5}}{5} i \]
The final solutions in the form \(a \pm bi\) are:
\[ 0 \pm \frac{2\sqrt{5}}{5} i \]
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