Solve Using the Quadratic Formula 6x^2=5x+6

$6 x^{2} = 5 x + 6$

Step 1

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

Subtract $5 x$ from both sides of the equation.

$6 x^{2} - 5 x = 6$

Subtract $6$ from both sides of the equation.

$6 x^{2} - 5 x - 6 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 6$ , $b = - 5$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (6 \cdot - 6)}}{2 \cdot 6}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 6 \cdot - 6}}{2 \cdot 6}$

Multiply $- 4 \cdot 6 \cdot - 6$ .

Multiply $- 4$ by $6$ .

$x = \frac{5 \pm \sqrt{25 - 24 \cdot - 6}}{2 \cdot 6}$

Multiply $- 24$ by $- 6$ .

$x = \frac{5 \pm \sqrt{25 + 144}}{2 \cdot 6}$

Add $25$ and $144$ .

$x = \frac{5 \pm \sqrt{169}}{2 \cdot 6}$

Rewrite $169$ as $13^{2}$ .

$x = \frac{5 \pm \sqrt{13^{2}}}{2 \cdot 6}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{5 \pm 13}{2 \cdot 6}$

Multiply $2$ by $6$ .

$x = \frac{5 \pm 13}{12}$

Step 5

The final answer is the combination of both solutions.

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