Find the Roots (Zeros) 8x^2=6x+9

$8 x^{2} = 6 x + 9$

Step 1

Subtract $6 x$ from both sides of the equation.

$8 x^{2} - 6 x = 9$

Step 2

Subtract $9$ from both sides of the equation.

$8 x^{2} - 6 x - 9 = 0$

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 $a \cdot c = 8 \cdot - 9 = - 72$ and whose sum is $b = - 6$ .

Factor $- 6$ out of $- 6 x$ .

$8 x^{2} - 6 x - 9 = 0$

Rewrite $- 6$ as $6$ plus $- 12$

$8 x^{2} + (6 - 12) x - 9 = 0$

Apply the distributive property.

$8 x^{2} + 6 x - 12 x - 9 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(8 x^{2} + 6 x) - 12 x - 9 = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $4 x + 3$ .

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

Step 4

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

$4 x + 3 = 0$

$2 x - 3 = 0$

Step 5

Set $4 x + 3$ equal to $0$ and solve for $x$ .

Set $4 x + 3$ equal to $0$ .

$4 x + 3 = 0$

Solve $4 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Divide each term in $4 x = - 3$ by $4$ and simplify.

Divide each term in $4 x = - 3$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 6

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

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

$2 x - 3 = 0$

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

Add $3$ to both sides of the equation.

$2 x = 3$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 7

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

$x = - \frac{3}{4}, \frac{3}{2}$

Step 8

Find the Roots (Zeros) 8x^2=6x+9

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