Find the Roots (Zeros) f(x)=-2x^2+16x-20

$f (x) = - 2 x^{2} + 16 x - 20$

Step 1

Set $- 2 x^{2} + 16 x - 20$ equal to $0$ .

$- 2 x^{2} + 16 x - 20 = 0$

Step 2

Solve for $x$ .

Factor $- 2$ out of $- 2 x^{2} + 16 x - 20$ .

Factor $- 2$ out of $- 2 x^{2}$ .

$- 2 x^{2} + 16 x - 20 = 0$

Factor $- 2$ out of $16 x$ .

$- 2 x^{2} - 2 (- 8 x) - 20 = 0$

Factor $- 2$ out of $- 20$ .

$- 2 x^{2} - 2 (- 8 x) - 2 \cdot 10 = 0$

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 8 x)$ .

$- 2 (x^{2} - 8 x) - 2 \cdot 10 = 0$

Factor $- 2$ out of $- 2 (x^{2} - 8 x) - 2 (10)$ .

$- 2 (x^{2} - 8 x + 10) = 0$

Divide each term in $- 2 (x^{2} - 8 x + 10) = 0$ by $- 2$ and simplify.

Divide each term in $- 2 (x^{2} - 8 x + 10) = 0$ by $- 2$ .

$\frac{- 2 (x^{2} - 8 x + 10)}{- 2} = \frac{0}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 (x^{2} - 8 x + 10)}{- 2} = \frac{0}{- 2}$

Divide $x^{2} - 8 x + 10$ by $1$ .

$x^{2} - 8 x + 10 = \frac{0}{- 2}$

Simplify the right side.

Divide $0$ by $- 2$ .

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 8$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 10)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 10$ .

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 10}}{2 \cdot 1}$

Multiply $- 4$ by $10$ .

$x = \frac{8 \pm \sqrt{64 - 40}}{2 \cdot 1}$

Subtract $40$ from $64$ .

$x = \frac{8 \pm \sqrt{24}}{2 \cdot 1}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$x = \frac{8 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{8 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{6}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{6}}{2}$

Simplify $\frac{8 \pm 2 \sqrt{6}}{2}$ .

$x = 4 \pm \sqrt{6}$

The final answer is the combination of both solutions.

$x = 4 + \sqrt{6}, 4 - \sqrt{6}$

$x = 4 \pm \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 4 \pm \sqrt{6}$

Decimal Form:

$x = 6.44948974 \dots, 1.55051025 \dots$

Step 4

Find the Roots (Zeros) f(x)=-2x^2+16x-20

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