Question

Use the quadratic formula to solve the quadratic equation
8
x
2
+
5
=
12
x
. Express its solutions in the form
a
±
b
i
.(1 point)

Answer 1

To solve the quadratic equation \( 8x^2 + 5 = 12x \), we first rearrange it into standard form. We do this by moving all terms to one side of the equation:

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

Now that we have it in the standard form \( ax^2 + bx + c = 0 \), we can identify the coefficients:
- \( a = 8 \)
- \( b = -12 \)
- \( c = 5 \)

Next, we use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Now we calculate the discriminant \( b^2 - 4ac \):

\[
b^2 = (-12)^2 = 144
\]
\[
4ac = 4 \times 8 \times 5 = 160
\]
\[
b^2 - 4ac = 144 - 160 = -16
\]

Since the discriminant is negative, the solutions will be complex numbers. We can now substitute the values into the quadratic formula:

\[
x = \frac{-(-12) \pm \sqrt{-16}}{2 \times 8}
\]
\[
x = \frac{12 \pm \sqrt{-16}}{16}
\]
\[
x = \frac{12 \pm 4i}{16}
\]

We can simplify this expression:

\[
x = \frac{12}{16} \pm \frac{4i}{16}
\]
\[
x = \frac{3}{4} \pm \frac{i}{4}
\]

Thus, the solutions of the quadratic equation \( 8x^2 + 5 = 12x \) in the form \( a \pm bi \) are:

\[
x = \frac{3}{4} \pm \frac{1}{4}i
\]