Solve by Factoring 12x^2-5x-25=0

$12 x^{2} - 5 x - 25 = 0$

Step 1

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 12 \cdot - 25 = - 300$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

$12 x^{2} - 5 x - 25 = 0$

Rewrite $- 5$ as $15$ plus $- 20$

$12 x^{2} + (15 - 20) x - 25 = 0$

Apply the distributive property.

$12 x^{2} + 15 x - 20 x - 25 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(12 x^{2} + 15 x) - 20 x - 25 = 0$

Factor out the greatest common factor (GCF) from each group.

$3 x (4 x + 5) - 5 (4 x + 5) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x + 5$ .

$(4 x + 5) (3 x - 5) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 x + 5 = 0$

$3 x - 5 = 0$

Step 3

Set $4 x + 5$ equal to $0$ and solve for $x$ .

Set $4 x + 5$ equal to $0$ .

$4 x + 5 = 0$

Solve $4 x + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$4 x = - 5$

Divide each term in $4 x = - 5$ by $4$ and simplify.

Divide each term in $4 x = - 5$ by $4$ .

$\frac{4 x}{4} = \frac{- 5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 5}{4}$

Divide $x$ by $1$ .

$x = \frac{- 5}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{4}$

Step 4

Set $3 x - 5$ equal to $0$ and solve for $x$ .

Set $3 x - 5$ equal to $0$ .

$3 x - 5 = 0$

Solve $3 x - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$3 x = 5$

Divide each term in $3 x = 5$ by $3$ and simplify.

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 5

The final solution is all the values that make $(4 x + 5) (3 x - 5) = 0$ true.

$x = - \frac{5}{4}, \frac{5}{3}$

Solve by Factoring 12x^2-5x-25=0

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