Question

Solve the equation 2x2=6x−5
2
𝑥
2
=
6
𝑥
−
5
. Write your conjugate pair solution(s) as two values separated by a ±
±
sign

Answer 1

To solve the equation \( 2x^2 = 6x - 5 \), we can first rearrange it into standard quadratic form by moving all terms to one side of the equation:

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

Next, we can use the quadratic formula to find the roots of the equation. The quadratic formula is given by:

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

Here, \( a = 2 \), \( b = -6 \), and \( c = 5 \). We can first calculate the discriminant (\( b^2 - 4ac \)):

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

Since the discriminant is negative, we will have complex solutions. Now we can apply the quadratic formula:

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

Simplifying further:

\[
\sqrt{-4} = 2i \quad \text{(where \( i \) is the imaginary unit)}
\]

Thus, we can express the solution as:

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

Simplifying this gives:

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

So, the two conjugate solution values are:

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

Thus, the final answer is:

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