Solve Using the Quadratic Formula 9x-4x^2=2

$9 x - 4 x^{2} = 2$

Step 1

Subtract $2$ from both sides of the equation.

$9 x - 4 x^{2} - 2 = 0$

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 $a = - 4$ , $b = 9$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (- 4 \cdot - 2)}}{2 \cdot - 4}$

Step 4

Simplify.

Simplify the numerator.

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot - 4 \cdot - 2}}{2 \cdot - 4}$

Multiply $- 4 \cdot - 4 \cdot - 2$ .

Multiply $- 4$ by $- 4$ .

$x = \frac{- 9 \pm \sqrt{81 + 16 \cdot - 2}}{2 \cdot - 4}$

Multiply $16$ by $- 2$ .

$x = \frac{- 9 \pm \sqrt{81 - 32}}{2 \cdot - 4}$

Subtract $32$ from $81$ .

$x = \frac{- 9 \pm \sqrt{49}}{2 \cdot - 4}$

Rewrite $49$ as $7^{2}$ .

$x = \frac{- 9 \pm \sqrt{7^{2}}}{2 \cdot - 4}$

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

$x = \frac{- 9 \pm 7}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$x = \frac{- 9 \pm 7}{- 8}$

Simplify $\frac{- 9 \pm 7}{- 8}$ .

$x = \frac{9 \pm 7}{8}$

Step 5

The final answer is the combination of both solutions.

$x = 2, \frac{1}{4}$