Finite Math Examples

$y = 16 x^{2} - 32 x + 20$

Step 1

Set $16 x^{2} - 32 x + 20$ equal to $0$ .

$16 x^{2} - 32 x + 20 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $4$ out of $16 x^{2} - 32 x + 20$ .

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Step 2.1.1

Factor $4$ out of $16 x^{2}$ .

$4 (4 x^{2}) - 32 x + 20 = 0$

Step 2.1.2

Factor $4$ out of $- 32 x$ .

$4 (4 x^{2}) + 4 (- 8 x) + 20 = 0$

Step 2.1.3

Factor $4$ out of $20$ .

$4 (4 x^{2}) + 4 (- 8 x) + 4 (5) = 0$

Step 2.1.4

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

$4 (4 x^{2} - 8 x) + 4 (5) = 0$

Step 2.1.5

Factor $4$ out of $4 (4 x^{2} - 8 x) + 4 (5)$ .

$4 (4 x^{2} - 8 x + 5) = 0$

Step 2.2

Divide each term in $4 (4 x^{2} - 8 x + 5) = 0$ by $4$ and simplify.

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Step 2.2.1

Divide each term in $4 (4 x^{2} - 8 x + 5) = 0$ by $4$ .

$\frac{4 (4 x^{2} - 8 x + 5)}{4} = \frac{0}{4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 (4 x^{2} - 8 x + 5)}{4} = \frac{0}{4}$

Step 2.2.2.1.2

Divide $4 x^{2} - 8 x + 5$ by $1$ .

$4 x^{2} - 8 x + 5 = \frac{0}{4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $4$ .

$4 x^{2} - 8 x + 5 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values $a = 4$ , $b = - 8$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (4 \cdot 5)}}{2 \cdot 4}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 4 \cdot 5}}{2 \cdot 4}$

Step 2.5.1.2

Multiply $- 4 \cdot 4 \cdot 5$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{8 \pm \sqrt{64 - 16 \cdot 5}}{2 \cdot 4}$

Step 2.5.1.2.2

Multiply $- 16$ by $5$ .

$x = \frac{8 \pm \sqrt{64 - 80}}{2 \cdot 4}$

Step 2.5.1.3

Subtract $80$ from $64$ .

$x = \frac{8 \pm \sqrt{- 16}}{2 \cdot 4}$

Step 2.5.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 4}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 4}$

Step 2.5.1.6

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

$x = \frac{8 \pm i \cdot \sqrt{16}}{2 \cdot 4}$

Step 2.5.1.7

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 4}$

Step 2.5.1.8

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

$x = \frac{8 \pm i \cdot 4}{2 \cdot 4}$

Step 2.5.1.9

Move $4$ to the left of $i$ .

$x = \frac{8 \pm 4 i}{2 \cdot 4}$

Step 2.5.2

Multiply $2$ by $4$ .

$x = \frac{8 \pm 4 i}{8}$

Step 2.5.3

Simplify $\frac{8 \pm 4 i}{8}$ .

$x = \frac{2 \pm i}{2}$

Step 2.6

The final answer is the combination of both solutions.

$x = 1 + \frac{i}{2}, 1 - \frac{i}{2}$

$x = \frac{2 \pm i}{2}$

Step 3