Finite Math Examples

$x^{2} - 2 x r + w^{2} = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - 2 r$ , and $c = w^{2}$ into the quadratic formula and solve for $x$ .

$\frac{2 r \pm \sqrt{{(- 2 r)}^{2} - 4 \cdot (1 \cdot w^{2})}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Rewrite $4 \cdot 1 \cdot w^{2}$ as ${(2 \cdot 1 w)}^{2}$ .

$x = \frac{2 r \pm \sqrt{{(- 2 r)}^{2} - {(2 \cdot (1 w))}^{2}}}{2 \cdot 1}$

Step 3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 2 r$ and $b = 2 \cdot 1 w$ .

$x = \frac{2 r \pm \sqrt{(- 2 r + 2 \cdot (1 w)) (- 2 r - (2 \cdot (1 w)))}}{2 \cdot 1}$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Factor $2$ out of $- 2 r + 2 \cdot 1 w$ .

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Step 3.1.3.1.1

Factor $2$ out of $- 2 r$ .

$x = \frac{2 r \pm \sqrt{(2 (- r) + 2 \cdot (1 w)) (- 2 r - (2 \cdot (1 w)))}}{2 \cdot 1}$

Step 3.1.3.1.2

Factor $2$ out of $2 \cdot 1 w$ .

$x = \frac{2 r \pm \sqrt{(2 (- r) + 2 (1 w)) (- 2 r - (2 \cdot (1 w)))}}{2 \cdot 1}$

Step 3.1.3.1.3

Factor $2$ out of $2 (- r) + 2 (1 w)$ .

$x = \frac{2 r \pm \sqrt{2 (- r + 1 w) (- 2 r - (2 \cdot (1 w)))}}{2 \cdot 1}$

Step 3.1.3.2

Multiply $w$ by $1$ .

$x = \frac{2 r \pm \sqrt{2 (- r + w) (- 2 r - (2 \cdot (1 w)))}}{2 \cdot 1}$

Step 3.1.3.3

Combine exponents.

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Step 3.1.3.3.1

Multiply $2$ by $1$ .

$x = \frac{2 r \pm \sqrt{2 (- r + w) (- 2 r - (2 w))}}{2 \cdot 1}$

Step 3.1.3.3.2

Multiply $2$ by $- 1$ .

$x = \frac{2 r \pm \sqrt{2 (- r + w) (- 2 r - 2 w)}}{2 \cdot 1}$

Step 3.1.4

Factor $2$ out of $- 2 r - 2 w$ .

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Step 3.1.4.1

Factor $2$ out of $- 2 r$ .

$x = \frac{2 r \pm \sqrt{2 (- r + w) (2 (- r) - 2 w)}}{2 \cdot 1}$

Step 3.1.4.2

Factor $2$ out of $- 2 w$ .

$x = \frac{2 r \pm \sqrt{2 (- r + w) (2 (- r) + 2 (- w))}}{2 \cdot 1}$

Step 3.1.4.3

Factor $2$ out of $2 (- r) + 2 (- w)$ .

$x = \frac{2 r \pm \sqrt{2 (- r + w) (2 (- r - w))}}{2 \cdot 1}$

$x = \frac{2 r \pm \sqrt{2 (- r + w) \cdot (2 (- r - w))}}{2 \cdot 1}$

Step 3.1.5

Multiply $2$ by $2$ .

$x = \frac{2 r \pm \sqrt{4 (- r + w) (- r - w)}}{2 \cdot 1}$

Step 3.1.6

Rewrite $4 (- r + w) (- r - w)$ as $2^{2} ((- r + w) (- r - w))$ .

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Step 3.1.6.1

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

$x = \frac{2 r \pm \sqrt{2^{2} (- r + w) (- r - w)}}{2 \cdot 1}$

Step 3.1.6.2

Add parentheses.

$x = \frac{2 r \pm \sqrt{2^{2} ((- r + w) (- r - w))}}{2 \cdot 1}$

Step 3.1.7

Pull terms out from under the radical.

$x = \frac{2 r \pm 2 \sqrt{(- r + w) (- r - w)}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{2 r \pm 2 \sqrt{(- r + w) (- r - w)}}{2}$

Step 3.3

Simplify $\frac{2 r \pm 2 \sqrt{(- r + w) (- r - w)}}{2}$ .

$x = r \pm \sqrt{(- r + w) (- r - w)}$

Step 4

The final answer is the combination of both solutions.

$x = r + \sqrt{(- r + w) (- r - w)}$

$x = r - \sqrt{(- r + w) (- r - w)}$