Question

Use the quadratic formula to solve the quadratic equation 8x2+5=12x . Express its solutions in the form a±bi .(1 point)

Answer 1

To solve the quadratic equation \(8x^2 + 5 = 12x\), we first rearrange it into standard form \(ax^2 + bx + c = 0\).

Starting with:
\[
8x^2 - 12x + 5 = 0
\]

Here, \(a = 8\), \(b = -12\), and \(c = 5\).

The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Now we will calculate \(b^2 - 4ac\):
\[
b^2 = (-12)^2 = 144
\]
\[
4ac = 4 \cdot 8 \cdot 5 = 160
\]
\[
b^2 - 4ac = 144 - 160 = -16
\]

Since \(b^2 - 4ac\) is negative, the equation has complex solutions. Now we can substitute back into the quadratic formula:
\[
x = \frac{-(-12) \pm \sqrt{-16}}{2 \cdot 8}
\]
\[
x = \frac{12 \pm \sqrt{-16}}{16}
\]
\[
x = \frac{12 \pm 4i}{16}
\]
\[
x = \frac{12}{16} \pm \frac{4i}{16}
\]
\[
x = \frac{3}{4} \pm \frac{i}{4}
\]

Thus, the solutions are:
\[
x = \frac{3}{4} + \frac{i}{4} \quad \text{and} \quad x = \frac{3}{4} - \frac{i}{4}
\]

In conclusion, the solutions in the form \(a \pm bi\) are:
\[
\frac{3}{4} \pm \frac{i}{4}
\]