Question

Use the quadratic formula to solve. Make sure to use ± when writing your solutions.
2x^2=6x-5

Answer 1

To solve the equation \(2x^2 = 6x - 5\) using the quadratic formula, we first need to rearrange the equation into standard form, which is \(ax^2 + bx + c = 0\).

Starting with the original equation:

\[
2x^2 - 6x + 5 = 0
\]

Here, we have:
- \(a = 2\)
- \(b = -6\)
- \(c = 5\)

The quadratic formula is given by:

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

Now, we will substitute the values of \(a\), \(b\), and \(c\) into the formula.

First, calculate the discriminant \(b^2 - 4ac\):

\[
b^2 = (-6)^2 = 36
\]
\[
4ac = 4 \cdot 2 \cdot 5 = 40
\]
\[
b^2 - 4ac = 36 - 40 = -4
\]

Since the discriminant is negative, this means we will have complex solutions.

Now we will apply the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{-4}}{2 \cdot 2}
\]
\[
x = \frac{6 \pm \sqrt{-4}}{4}
\]

Recall that \(\sqrt{-4}\) can be expressed as \(2i\):

\[
x = \frac{6 \pm 2i}{4}
\]

Now simplify:

\[
x = \frac{6}{4} \pm \frac{2i}{4}
\]
\[
x = \frac{3}{2} \pm \frac{i}{2}
\]

Thus, the solutions to the equation \(2x^2 = 6x - 5\) are:

\[
x = \frac{3}{2} + \frac{i}{2} \quad \text{and} \quad x = \frac{3}{2} - \frac{i}{2}
\]

So, the final answers can be written as:

\[
x = \frac{3}{2} \pm \frac{i}{2}
\]