To solve the quadratic equation \( 10x^2 + 5x = 5x - 8 \), we first need to rearrange the equation to standard form, \( ax^2 + bx + c = 0 \).
- Start by subtracting \( 5x \) from both sides:
\[ 10x^2 + 5x - 5x = -8 \]
This simplifies to:
\[ 10x^2 = -8 \]
- Next, add \( 8 \) to both sides:
\[ 10x^2 + 8 = 0 \]
- Move \( 8 \) to the right side:
\[ 10x^2 = -8 \]
- Divide each term by \( 10 \) to simplify:
\[ x^2 = -\frac{8}{10} = -\frac{4}{5} \]
- Now, we rearrange this to standard form of a quadratic equation:
\[ x^2 + \frac{4}{5} = 0 \]
The equation is:
\[ x^2 + 0x + \frac{4}{5} = 0 \]
Here, \( a = 1 \), \( b = 0 \), and \( c = \frac{4}{5} \).
Now, we will use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
- Calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 0^2 - 4 \cdot 1 \cdot \frac{4}{5} = 0 - \frac{16}{5} = -\frac{16}{5} \]
- Now substitute \( a \), \( b \), and the discriminant into the quadratic formula:
\[ x = \frac{-0 \pm \sqrt{-\frac{16}{5}}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{\pm \sqrt{-\frac{16}{5}}}{2} \]
- The square root of a negative number can be expressed in terms of \( i \):
\[ \sqrt{-\frac{16}{5}} = \sqrt{16} \cdot \sqrt{-1} / \sqrt{5} = 4i/\sqrt{5} \]
Substituting back into the equation for \( x \):
\[ x = \frac{\pm \frac{4i}{\sqrt{5}}}{2} \]
\[ x = \pm \frac{2i}{\sqrt{5}} \]
- We can write \( \frac{2i}{\sqrt{5}} \) in a form involving \( a \) and \( b \):
Let \( a = 0 \) and \( b = \frac{2}{\sqrt{5}} \):
Thus, the solutions can be expressed as:
\[ x = 0 \pm \frac{2}{\sqrt{5}} i \]
In conclusion, the solutions of the quadratic equation \( 10x^2 + 5x = 5x - 8 \) are:
\[ x = 0 \pm \frac{2}{\sqrt{5}} i \]
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