Solve by Factoring 10x^2-13x-17=-8

$10 x^{2} - 13 x - 17 = - 8$

Step 1

Add $8$ to both sides of the equation.

$10 x^{2} - 13 x - 17 + 8 = 0$

Step 2

Add $- 17$ and $8$ .

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

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 $a \cdot c = 10 \cdot - 9 = - 90$ and whose sum is $b = - 13$ .

Factor $- 13$ out of $- 13 x$ .

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

Rewrite $- 13$ as $5$ plus $- 18$

$10 x^{2} + (5 - 18) x - 9 = 0$

Apply the distributive property.

$10 x^{2} + 5 x - 18 x - 9 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(10 x^{2} + 5 x) - 18 x - 9 = 0$

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

$5 x (2 x + 1) - 9 (2 x + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$(2 x + 1) (5 x - 9) = 0$

Step 4

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

$2 x + 1 = 0$

$5 x - 9 = 0$

Step 5

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve $2 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term in $2 x = - 1$ by $2$ and simplify.

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 6

Set $5 x - 9$ equal to $0$ and solve for $x$ .

Set $5 x - 9$ equal to $0$ .

$5 x - 9 = 0$

Solve $5 x - 9 = 0$ for $x$ .

Add $9$ to both sides of the equation.

$5 x = 9$

Divide each term in $5 x = 9$ by $5$ and simplify.

Divide each term in $5 x = 9$ by $5$ .

$\frac{5 x}{5} = \frac{9}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{9}{5}$

Divide $x$ by $1$ .

$x = \frac{9}{5}$

Step 7

The final solution is all the values that make $(2 x + 1) (5 x - 9) = 0$ true.

$x = - \frac{1}{2}, \frac{9}{5}$

Solve by Factoring 10x^2-13x-17=-8

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