Question

question
solve the quadratic by factoring.
$5x^2 - 6x - 2 = -9x$

Explanation:

Step1: Rearrange the equation

First, we need to get all terms on one side to set the equation to zero. Add \(9x\) to both sides of the equation \(5x^{2}-6x - 2=-9x\).

$$
LATEXBLOCK0
$$

Step2: Factor the quadratic

We need to factor \(5x^{2}+3x - 2\). We look for two numbers that multiply to \(5\times(-2)=-10\) and add up to \(3\). The numbers are \(5\) and \(-2\) since \(5\times(-2)=-10\) and \(5+(-2) = 3\).
Rewrite the middle term using these numbers:

$$ 5x^{2}+5x-2x - 2=0 $$

Group the terms:

$$ (5x^{2}+5x)+(-2x - 2)=0 $$

Factor out the common factors from each group:

$$ 5x(x + 1)-2(x + 1)=0 $$

Now, factor out \((x + 1)\):

$$ (5x - 2)(x + 1)=0 $$

Step3: Solve for \(x\)

Set each factor equal to zero and solve for \(x\).
For \(5x-2 = 0\):

$$
LATEXBLOCK1
$$

For \(x + 1=0\):

$$
LATEXBLOCK2
$$

Answer:

\(x=-1\) or \(x = \frac{2}{5}\)