Question

solve the equation by factoring.
$6x^2 - 4 = 5x$
$x = \square$
(use a comma to separate answers as needed. sim

Explanation:

Step1: Rearrange the equation

First, we need to rewrite the equation \(6x^2 - 4 = 5x\) in standard quadratic form \(ax^2 + bx + c = 0\). Subtract \(5x\) from both sides:
\(6x^2 - 5x - 4 = 0\)

Step2: Factor the quadratic

We need to find two numbers that multiply to \(6\times(-4)= -24\) and add up to \(-5\). The numbers are \(-8\) and \(3\) because \(-8\times3 = -24\) and \(-8 + 3 = -5\).
Now, we rewrite the middle term using these numbers:
\(6x^2 + 3x - 8x - 4 = 0\)
Factor by grouping:
Group the first two terms and the last two terms:
\((6x^2 + 3x) + (-8x - 4) = 0\)
Factor out the greatest common factor from each group:
\(3x(2x + 1) - 4(2x + 1) = 0\)
Now, factor out the common binomial factor \((2x + 1)\):
\((3x - 4)(2x + 1) = 0\)

Step3: Solve for x

Set each factor equal to zero and solve for \(x\):

1. \(3x - 4 = 0\)

Add \(4\) to both sides: \(3x = 4\)
Divide both sides by \(3\): \(x = \frac{4}{3}\)

1. \(2x + 1 = 0\)

Subtract \(1\) from both sides: \(2x = -1\)
Divide both sides by \(2\): \(x = -\frac{1}{2}\)

Answer:

\(\frac{4}{3}, -\frac{1}{2}\)