Algebra Examples

$8 q (q - 2) + 10 = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$8 q \cdot q + 8 q \cdot - 2 + 10 = 0$

Step 1.1.2

Multiply $q$ by $q$ by adding the exponents.

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Step 1.1.2.1

Move $q$ .

$8 (q \cdot q) + 8 q \cdot - 2 + 10 = 0$

Step 1.1.2.2

Multiply $q$ by $q$ .

$8 q^{2} + 8 q \cdot - 2 + 10 = 0$

Step 1.1.3

Multiply $- 2$ by $8$ .

$8 q^{2} - 16 q + 10 = 0$

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 $a = 8$ , $b = - 16$ , and $c = 10$ into the quadratic formula and solve for $q$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (8 \cdot 10)}}{2 \cdot 8}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 16$ to the power of $2$ .

$q = \frac{16 \pm \sqrt{256 - 4 \cdot 8 \cdot 10}}{2 \cdot 8}$

Step 4.1.2

Multiply $- 4 \cdot 8 \cdot 10$ .

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Step 4.1.2.1

Multiply $- 4$ by $8$ .

$q = \frac{16 \pm \sqrt{256 - 32 \cdot 10}}{2 \cdot 8}$

Step 4.1.2.2

Multiply $- 32$ by $10$ .

$q = \frac{16 \pm \sqrt{256 - 320}}{2 \cdot 8}$

Step 4.1.3

Subtract $320$ from $256$ .

$q = \frac{16 \pm \sqrt{- 64}}{2 \cdot 8}$

Step 4.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$q = \frac{16 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 8}$

Step 4.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$q = \frac{16 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 8}$

Step 4.1.6

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

$q = \frac{16 \pm i \cdot \sqrt{64}}{2 \cdot 8}$

Step 4.1.7

Rewrite $64$ as $8^{2}$ .

$q = \frac{16 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 8}$

Step 4.1.8

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

$q = \frac{16 \pm i \cdot 8}{2 \cdot 8}$

Step 4.1.9

Move $8$ to the left of $i$ .

$q = \frac{16 \pm 8 i}{2 \cdot 8}$

Step 4.2

Multiply $2$ by $8$ .

$q = \frac{16 \pm 8 i}{16}$

Step 4.3

Simplify $\frac{16 \pm 8 i}{16}$ .

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