Algebra Examples

$w (7 w - 10) + 4 = 0$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$w (7 w) + w \cdot - 10 + 4 = 0$

Step 1.2

Rewrite using the commutative property of multiplication.

$7 w \cdot w + w \cdot - 10 + 4 = 0$

Step 1.3

Move $- 10$ to the left of $w$ .

$7 w \cdot w - 10 \cdot w + 4 = 0$

Step 1.4

Multiply $w$ by $w$ by adding the exponents.

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Step 1.4.1

Move $w$ .

$7 (w \cdot w) - 10 \cdot w + 4 = 0$

Step 1.4.2

Multiply $w$ by $w$ .

$7 w^{2} - 10 \cdot w + 4 = 0$

$7 w^{2} - 10 w + 4 = 0$

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 $a = 7$ , $b = - 10$ , and $c = 4$ into the quadratic formula and solve for $w$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (7 \cdot 4)}}{2 \cdot 7}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 10$ to the power of $2$ .

$w = \frac{10 \pm \sqrt{100 - 4 \cdot 7 \cdot 4}}{2 \cdot 7}$

Step 4.1.2

Multiply $- 4 \cdot 7 \cdot 4$ .

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Step 4.1.2.1

Multiply $- 4$ by $7$ .

$w = \frac{10 \pm \sqrt{100 - 28 \cdot 4}}{2 \cdot 7}$

Step 4.1.2.2

Multiply $- 28$ by $4$ .

$w = \frac{10 \pm \sqrt{100 - 112}}{2 \cdot 7}$

Step 4.1.3

Subtract $112$ from $100$ .

$w = \frac{10 \pm \sqrt{- 12}}{2 \cdot 7}$

Step 4.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$w = \frac{10 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 7}$

Step 4.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$w = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 7}$

Step 4.1.6

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

$w = \frac{10 \pm i \cdot \sqrt{12}}{2 \cdot 7}$

Step 4.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.1.7.1

Factor $4$ out of $12$ .

$w = \frac{10 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 7}$

Step 4.1.7.2

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

$w = \frac{10 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 7}$

Step 4.1.8

Pull terms out from under the radical.

$w = \frac{10 \pm i \cdot (2 \sqrt{3})}{2 \cdot 7}$

Step 4.1.9

Move $2$ to the left of $i$ .

$w = \frac{10 \pm 2 i \sqrt{3}}{2 \cdot 7}$

Step 4.2

Multiply $2$ by $7$ .

$w = \frac{10 \pm 2 i \sqrt{3}}{14}$

Step 4.3

Simplify $\frac{10 \pm 2 i \sqrt{3}}{14}$ .

$w = \frac{5 \pm i \sqrt{3}}{7}$

Step 5

The final answer is the combination of both solutions.

$w = \frac{5 + i \sqrt{3}}{7}, \frac{5 - i \sqrt{3}}{7}$