Solve by Factoring 9x^2-13x=8x-10

$9 x^{2} - 13 x = 8 x - 10$

Step 1

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

Subtract $8 x$ from both sides of the equation.

$9 x^{2} - 13 x - 8 x = - 10$

Add $10$ to both sides of the equation.

$9 x^{2} - 13 x - 8 x + 10 = 0$

Step 2

Subtract $8 x$ from $- 13 x$ .

$9 x^{2} - 21 x + 10 = 0$

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 $a \cdot c = 9 \cdot 10 = 90$ and whose sum is $b = - 21$ .

Factor $- 21$ out of $- 21 x$ .

$9 x^{2} - 21 x + 10 = 0$

Rewrite $- 21$ as $- 6$ plus $- 15$

$9 x^{2} + (- 6 - 15) x + 10 = 0$

Apply the distributive property.

$9 x^{2} - 6 x - 15 x + 10 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(9 x^{2} - 6 x) - 15 x + 10 = 0$

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

$3 x (3 x - 2) - 5 (3 x - 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$(3 x - 2) (3 x - 5) = 0$

Step 4

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

$3 x - 2 = 0$

$3 x - 5 = 0$

Step 5

Set $3 x - 2$ equal to $0$ and solve for $x$ .

Set $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Solve $3 x - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$3 x = 2$

Divide each term in $3 x = 2$ by $3$ and simplify.

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 6

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 7

The final solution is all the values that make $(3 x - 2) (3 x - 5) = 0$ true.

$x = \frac{2}{3}, \frac{5}{3}$

Solve by Factoring 9x^2-13x=8x-10

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