Solve Using the Quadratic Formula 16x^2-24x+9=0

$16 x^{2} - 24 x + 9 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 16$ , $b = - 24$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (16 \cdot 9)}}{2 \cdot 16}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 16 \cdot 9}}{2 \cdot 16}$

Multiply $- 4 \cdot 16 \cdot 9$ .

Multiply $- 4$ by $16$ .

$x = \frac{24 \pm \sqrt{576 - 64 \cdot 9}}{2 \cdot 16}$

Multiply $- 64$ by $9$ .

$x = \frac{24 \pm \sqrt{576 - 576}}{2 \cdot 16}$

Subtract $576$ from $576$ .

$x = \frac{24 \pm \sqrt{0}}{2 \cdot 16}$

Rewrite $0$ as $0^{2}$ .

$x = \frac{24 \pm \sqrt{0^{2}}}{2 \cdot 16}$

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

$x = \frac{24 \pm 0}{2 \cdot 16}$

$24$ plus or minus $0$ is $24$ .

$x = \frac{24}{2 \cdot 16}$

Multiply $2$ by $16$ .

$x = \frac{24}{32}$

Cancel the common factor of $24$ and $32$ .

Factor $8$ out of $24$ .

$x = \frac{8 (3)}{32}$

Cancel the common factors.

Factor $8$ out of $32$ .

$x = \frac{8 \cdot 3}{8 \cdot 4}$

Cancel the common factor.

$x = \frac{8 \cdot 3}{8 \cdot 4}$

Rewrite the expression.

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

Step 4

The final answer is the combination of both solutions.

$x = \frac{3}{4}$ Double roots