Find the Roots (Zeros) f(x)=-4x^2+10x-8

$f (x) = - 4 x^{2} + 10 x - 8$

Step 1

Set $- 4 x^{2} + 10 x - 8$ equal to $0$ .

$- 4 x^{2} + 10 x - 8 = 0$

Step 2

Solve for $x$ .

Factor $- 2$ out of $- 4 x^{2} + 10 x - 8$ .

Factor $- 2$ out of $- 4 x^{2}$ .

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

Factor $- 2$ out of $10 x$ .

$- 2 (2 x^{2}) - 2 (- 5 x) - 8 = 0$

Factor $- 2$ out of $- 8$ .

$- 2 (2 x^{2}) - 2 (- 5 x) - 2 \cdot 4 = 0$

Factor $- 2$ out of $- 2 (2 x^{2}) - 2 (- 5 x)$ .

$- 2 (2 x^{2} - 5 x) - 2 \cdot 4 = 0$

Factor $- 2$ out of $- 2 (2 x^{2} - 5 x) - 2 (4)$ .

$- 2 (2 x^{2} - 5 x + 4) = 0$

Divide each term in $- 2 (2 x^{2} - 5 x + 4) = 0$ by $- 2$ and simplify.

Divide each term in $- 2 (2 x^{2} - 5 x + 4) = 0$ by $- 2$ .

$\frac{- 2 (2 x^{2} - 5 x + 4)}{- 2} = \frac{0}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 (2 x^{2} - 5 x + 4)}{- 2} = \frac{0}{- 2}$

Divide $2 x^{2} - 5 x + 4$ by $1$ .

$2 x^{2} - 5 x + 4 = \frac{0}{- 2}$

Simplify the right side.

Divide $0$ by $- 2$ .

$2 x^{2} - 5 x + 4 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 2$ , $b = - 5$ , and $c = 4$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 8 \cdot 4}}{2 \cdot 2}$

Multiply $- 8$ by $4$ .

$x = \frac{5 \pm \sqrt{25 - 32}}{2 \cdot 2}$

Subtract $32$ from $25$ .

$x = \frac{5 \pm \sqrt{- 7}}{2 \cdot 2}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 2}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{7}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{5 \pm i \sqrt{7}}{4}$

The final answer is the combination of both solutions.

$x = \frac{5 + i \sqrt{7}}{4}, \frac{5 - i \sqrt{7}}{4}$

$x = \frac{5 \pm i \sqrt{7}}{4}$

Step 3

Find the Roots (Zeros) f(x)=-4x^2+10x-8

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