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 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:

\[
\frac{3}{4} + \frac{i}{4} \quad \text{and} \quad \frac{3}{4} - \frac{i}{4}
\]