Solve by Factoring 24x^2-46x=18

$24 x^{2} - 46 x = 18$

Step 1

Subtract $18$ from both sides of the equation.

$24 x^{2} - 46 x - 18 = 0$

Step 2

Factor $2$ out of $24 x^{2} - 46 x - 18$ .

Factor $2$ out of $24 x^{2}$ .

$2 (12 x^{2}) - 46 x - 18 = 0$

Factor $2$ out of $- 46 x$ .

$2 (12 x^{2}) + 2 (- 23 x) - 18 = 0$

Factor $2$ out of $- 18$ .

$2 (12 x^{2}) + 2 (- 23 x) + 2 (- 9) = 0$

Factor $2$ out of $2 (12 x^{2}) + 2 (- 23 x)$ .

$2 (12 x^{2} - 23 x) + 2 (- 9) = 0$

Factor $2$ out of $2 (12 x^{2} - 23 x) + 2 (- 9)$ .

$2 (12 x^{2} - 23 x - 9) = 0$

Step 3

Factor.

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 $a \cdot c = 12 \cdot - 9 = - 108$ and whose sum is $b = - 23$ .

Factor $- 23$ out of $- 23 x$ .

$2 (12 x^{2} - 23 x - 9) = 0$

Rewrite $- 23$ as $4$ plus $- 27$

$2 (12 x^{2} + (4 - 27) x - 9) = 0$

Apply the distributive property.

$2 (12 x^{2} + 4 x - 27 x - 9) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 ((12 x^{2} + 4 x) - 27 x - 9) = 0$

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

$2 (4 x (3 x + 1) - 9 (3 x + 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$2 ((3 x + 1) (4 x - 9)) = 0$

Remove unnecessary parentheses.

$2 (3 x + 1) (4 x - 9) = 0$

Step 4

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

$3 x + 1 = 0$

$4 x - 9 = 0$

Step 5

Set $3 x + 1$ equal to $0$ and solve for $x$ .

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Solve $3 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Divide each term in $3 x = - 1$ by $3$ and simplify.

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 6

Set $4 x - 9$ equal to $0$ and solve for $x$ .

Set $4 x - 9$ equal to $0$ .

$4 x - 9 = 0$

Solve $4 x - 9 = 0$ for $x$ .

Add $9$ to both sides of the equation.

$4 x = 9$

Divide each term in $4 x = 9$ by $4$ and simplify.

Divide each term in $4 x = 9$ by $4$ .

$\frac{4 x}{4} = \frac{9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{9}{4}$

Divide $x$ by $1$ .

$x = \frac{9}{4}$

Step 7

The final solution is all the values that make $2 (3 x + 1) (4 x - 9) = 0$ true.

$x = - \frac{1}{3}, \frac{9}{4}$

Solve by Factoring 24x^2-46x=18

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