Basic Math Examples

$w - 7^{2} = 2 w^{2} - 6 w + 56$

Step 1

Since $w$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 w^{2} - 6 w + 56 = w - 7^{2}$

Step 2

Simplify each term.

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Step 2.1

Raise $7$ to the power of $2$ .

$2 w^{2} - 6 w + 56 = w - 1 \cdot 49$

Step 2.2

Multiply $- 1$ by $49$ .

$2 w^{2} - 6 w + 56 = w - 49$

Step 3

Move all terms containing $w$ to the left side of the equation.

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Step 3.1

Subtract $w$ from both sides of the equation.

$2 w^{2} - 6 w + 56 - w = - 49$

Step 3.2

Subtract $w$ from $- 6 w$ .

$2 w^{2} - 7 w + 56 = - 49$

Step 4

Move all terms to the left side of the equation and simplify.

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Step 4.1

Add $49$ to both sides of the equation.

$2 w^{2} - 7 w + 56 + 49 = 0$

Step 4.2

Add $56$ and $49$ .

$2 w^{2} - 7 w + 105 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 2$ , $b = - 7$ , and $c = 105$ into the quadratic formula and solve for $w$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (2 \cdot 105)}}{2 \cdot 2}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 7$ to the power of $2$ .

$w = \frac{7 \pm \sqrt{49 - 4 \cdot 2 \cdot 105}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot 105$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$w = \frac{7 \pm \sqrt{49 - 8 \cdot 105}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $105$ .

$w = \frac{7 \pm \sqrt{49 - 840}}{2 \cdot 2}$

Step 7.1.3

Subtract $840$ from $49$ .

$w = \frac{7 \pm \sqrt{- 791}}{2 \cdot 2}$

Step 7.1.4

Rewrite $- 791$ as $- 1 (791)$ .

$w = \frac{7 \pm \sqrt{- 1 \cdot 791}}{2 \cdot 2}$

Step 7.1.5

Rewrite $\sqrt{- 1 (791)}$ as $\sqrt{- 1} \cdot \sqrt{791}$ .

$w = \frac{7 \pm \sqrt{- 1} \cdot \sqrt{791}}{2 \cdot 2}$

Step 7.1.6

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

$w = \frac{7 \pm i \sqrt{791}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$w = \frac{7 \pm i \sqrt{791}}{4}$

Step 8

The final answer is the combination of both solutions.

$w = \frac{7 + i \sqrt{791}}{4}, \frac{7 - i \sqrt{791}}{4}$