Question

find the exact solutions to $3x^2 = 5x - 1$ by using the quadratic formula.

$\frac{-5 \pm \sqrt{13}}{6}$

$\frac{5 \pm \sqrt{13}}{6}$

$\frac{5 \pm \sqrt{37}}{6}$

$5 \pm \frac{\sqrt{13}}{6}$

Explanation:

Step1: Rewrite in standard form

First, rewrite the equation \(3x^2 = 5x - 1\) in standard quadratic form \(ax^2 + bx + c = 0\). Subtract \(5x\) and add \(1\) to both sides:
\(3x^2 - 5x + 1 = 0\)
Here, \(a = 3\), \(b = -5\), \(c = 1\).

Step2: Apply Quadratic Formula

The Quadratic Formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 3\), \(b = -5\), \(c = 1\) into the formula:
First, calculate \(-b\): \(-b=-(-5) = 5\)
Then, calculate the discriminant \(b^2 - 4ac\): \((-5)^2 - 4(3)(1)=25 - 12 = 13\)
Now, substitute into the formula: \(x=\frac{5\pm\sqrt{13}}{2(3)}=\frac{5\pm\sqrt{13}}{6}\)

Answer:

\(\boldsymbol{\frac{5\pm\sqrt{13}}{6}}\) (corresponding to the option " \(\frac{5\pm\sqrt{13}}{6}\) ")