To solve the quadratic equation \( 8x^2 + 5 = 12x \), we first rearrange the equation to standard form:
\[ 8x^2 - 12x + 5 = 0 \]
Now, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, the coefficients are:
- \( a = 8 \)
- \( b = -12 \)
- \( c = 5 \)
Now, we calculate \( b^2 - 4ac \):
\[ b^2 = (-12)^2 = 144 \] \[ 4ac = 4 \cdot 8 \cdot 5 = 160 \] \[ b^2 - 4ac = 144 - 160 = -16 \]
Since \( b^2 - 4ac = -16 \), we have a negative discriminant, which means the solutions will be complex.
Next, substitute the values into the quadratic formula:
\[ x = \frac{-(-12) \pm \sqrt{-16}}{2 \cdot 8} \] \[ x = \frac{12 \pm \sqrt{-16}}{16} \] \[ x = \frac{12 \pm 4i}{16} \]
Now simplify the expression:
\[ x = \frac{12}{16} \pm \frac{4i}{16} \] \[ x = \frac{3}{4} \pm \frac{1}{4}i \]
Thus, the solutions to the equation \( 8x^2 - 12x + 5 = 0 \) are:
\[ x = \frac{3}{4} \pm \frac{1}{4}i \]
Now, looking for the solution in the provided options, we see that the correct expression corresponds to:
\[ \frac{3}{4} \pm \frac{1}{4}i \]
This matches with option A:
\[ \frac{3}{4} \pm \frac{1}{4}i \]
So the final answer is:
A: \(\frac{3}{4} \pm \frac{1}{4}i\)