Bot GPT 4-o mini To solve the quadratic equation \( 10x^2 + 5x = 5x - 8 \), first, we need to rearrange it into standard form \( ax^2 + bx + c = 0 \).
Starting with the original equation:
\[ 10x^2 + 5x = 5x - 8 \]
Subtract \( 5x \) from both sides:
\[ 10x^2 + 5x - 5x = -8 \]
This simplifies to:
\[ 10x^2 = -8 \]
Now, add \( 8 \) to both sides to set the equation to zero:
\[ 10x^2 + 8 = 0 \]
Next, we can rewrite this as:
\[ 10x^2 + 0x + 8 = 0 \]
Here, we have \( a = 10 \), \( b = 0 \), and \( c = 8 \).
Now we can apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting in the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-0 \pm \sqrt{0^2 - 4 \cdot 10 \cdot 8}}{2 \cdot 10} \]
Calculating under the square root:
\[ x = \frac{0 \pm \sqrt{0 - 320}}{20} \]
This simplifies to:
\[ x = \frac{\pm \sqrt{-320}}{20} \]
Noting that \( \sqrt{-320} = \sqrt{320}i \):
\[ \sqrt{320} = \sqrt{64 \cdot 5} = 8\sqrt{5} \]
Thus, we have:
\[ x = \frac{\pm 8\sqrt{5}i}{20} \]
This can be simplified:
\[ x = \frac{\pm 2\sqrt{5}i}{5} \]
Finally, we can write the solutions in the form \( a \pm bi \):
\[ x = 0 \pm \frac{2\sqrt{5}}{5}i \]
Therefore, the solutions to the quadratic equation are:
\[ x = 0 \pm \frac{2\sqrt{5}}{5}i \]
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