To solve the equation \( 6x + 5 = 4x^2 - 8 \), we first rearrange it to one side to set it to zero:
\[ 4x^2 - 6x - 8 - 5 = 0 \]
This simplifies to:
\[ 4x^2 - 6x - 13 = 0 \]
Next, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4 \), \( b = -6 \), and \( c = -13 \).
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (-6)^2 = 36 \] \[ 4ac = 4 \cdot 4 \cdot (-13) = -208 \]
Thus, the discriminant is:
\[ b^2 - 4ac = 36 + 208 = 244 \]
Now we can find the solutions using the quadratic formula:
\[ x = \frac{-(-6) \pm \sqrt{244}}{2 \cdot 4} \]
\[ x = \frac{6 \pm \sqrt{244}}{8} \]
Now we simplify \( \sqrt{244} \):
\[ \sqrt{244} = \sqrt{4 \cdot 61} = 2\sqrt{61} \]
Substituting this back into the formula, we have:
\[ x = \frac{6 \pm 2\sqrt{61}}{8} \]
We can simplify this expression:
\[ x = \frac{6}{8} \pm \frac{2\sqrt{61}}{8} \]
\[ x = \frac{3}{4} \pm \frac{\sqrt{61}}{4} \]
Thus, the solutions for the equation \( 6x + 5 = 4x^2 - 8 \) are:
\[ x = \frac{3 + \sqrt{61}}{4} \quad \text{and} \quad x = \frac{3 - \sqrt{61}}{4} \]
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