Solve by Factoring 7x^2-12x=36+7x

$7 x^{2} - 12 x = 36 + 7 x$

Step 1

Move all the expressions to the left side of the equation.

Subtract $36$ from both sides of the equation.

$7 x^{2} - 12 x - 36 = 7 x$

Subtract $7 x$ from both sides of the equation.

$7 x^{2} - 12 x - 36 - 7 x = 0$

Step 2

Subtract $7 x$ from $- 12 x$ .

$7 x^{2} - 19 x - 36 = 0$

Step 3

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 $a \cdot c = 7 \cdot - 36 = - 252$ and whose sum is $b = - 19$ .

Factor $- 19$ out of $- 19 x$ .

$7 x^{2} - 19 x - 36 = 0$

Rewrite $- 19$ as $9$ plus $- 28$

$7 x^{2} + (9 - 28) x - 36 = 0$

Apply the distributive property.

$7 x^{2} + 9 x - 28 x - 36 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(7 x^{2} + 9 x) - 28 x - 36 = 0$

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

$x (7 x + 9) - 4 (7 x + 9) = 0$

Factor the polynomial by factoring out the greatest common factor, $7 x + 9$ .

$(7 x + 9) (x - 4) = 0$

Step 4

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

$7 x + 9 = 0$

$x - 4 = 0$

Step 5

Set $7 x + 9$ equal to $0$ and solve for $x$ .

Set $7 x + 9$ equal to $0$ .

$7 x + 9 = 0$

Solve $7 x + 9 = 0$ for $x$ .

Subtract $9$ from both sides of the equation.

$7 x = - 9$

Divide each term in $7 x = - 9$ by $7$ and simplify.

Divide each term in $7 x = - 9$ by $7$ .

$\frac{7 x}{7} = \frac{- 9}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 9}{7}$

Divide $x$ by $1$ .

$x = \frac{- 9}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{9}{7}$

Step 6

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 7

The final solution is all the values that make $(7 x + 9) (x - 4) = 0$ true.

$x = - \frac{9}{7}, 4$

Solve by Factoring 7x^2-12x=36+7x

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