To solve the quadratic equation \( 8x^2 + 5 = 12x \), we first rearrange it into standard form:
\[ 8x^2 - 12x + 5 = 0 \]
In the standard form \( ax^2 + bx + c = 0 \), we have:
- \( a = 8 \)
- \( b = -12 \)
- \( c = 5 \)
Next, we apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Calculating the discriminant \( b^2 - 4ac \):
\[ b^2 = (-12)^2 = 144 \] \[ 4ac = 4 \cdot 8 \cdot 5 = 160 \] \[ b^2 - 4ac = 144 - 160 = -16 \]
Since the discriminant is negative, we will have complex solutions. Now, we substitute \( a \), \( b \), and the discriminant into the quadratic formula:
\[ x = \frac{-(-12) \pm \sqrt{-16}}{2 \cdot 8} \] \[ x = \frac{12 \pm \sqrt{-16}}{16} \]
We can simplify \( \sqrt{-16} \):
\[ \sqrt{-16} = 4i \]
Substituting this back into the formula gives us:
\[ x = \frac{12 \pm 4i}{16} \]
We can simplify this expression further:
\[ x = \frac{12}{16} \pm \frac{4i}{16} \] \[ x = \frac{3}{4} \pm \frac{i}{4} \]
Thus, the solutions to the quadratic equation \( 8x^2 + 5 = 12x \) are:
\[ x = \frac{3}{4} \pm \frac{1}{4}i \]
Final answer:
\[ x = \frac{3}{4} \pm \frac{1}{4}i \]