To solve the quadratic equation \(8x^2 + 5 = 12x\), we first rearrange the equation into standard form:
\[ 8x^2 - 12x + 5 = 0 \]
Now, we can identify the coefficients \(a\), \(b\), and \(c\) from the standard form \(ax^2 + bx + c = 0\):
- \(a = 8\)
- \(b = -12\)
- \(c = 5\)
Next, we will use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
We first calculate the discriminant:
\[ b^2 - 4ac = (-12)^2 - 4 \cdot 8 \cdot 5 \]
Calculating each part:
\[ (-12)^2 = 144 \] \[ 4 \cdot 8 \cdot 5 = 160 \]
Now, we substitute these values into the discriminant:
\[ b^2 - 4ac = 144 - 160 = -16 \]
The discriminant is negative, which means the solutions will be complex numbers. Now we substitute back into the quadratic formula:
\[ x = \frac{-(-12) \pm \sqrt{-16}}{2 \cdot 8} \] \[ x = \frac{12 \pm \sqrt{-16}}{16} \]
Since \(\sqrt{-16} = 4i\):
\[ x = \frac{12 \pm 4i}{16} \]
We can simplify this:
\[ x = \frac{12}{16} \pm \frac{4i}{16} \] \[ x = \frac{3}{4} \pm \frac{i}{4} \]
Thus, the solutions to the quadratic equation \(8x^2 - 12x + 5 = 0\) are:
\[ x = \frac{3}{4} \pm \frac{i}{4} \]
These can be expressed in the form \(a \pm bi\) as required. The final answers are:
\[ \frac{3}{4} + \frac{i}{4} \quad \text{and} \quad \frac{3}{4} - \frac{i}{4} \]
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