Finite Math Examples

a^{2} + b^{2} = 484

$a^{2} + b^{2} = 484$

Step 1

Subtract

484

$484$ from both sides of the equation.

a^{2} + b^{2} - 484 = 0

$a^{2} + b^{2} - 484 = 0$

Step 2

Factor

a^{2} + b^{2} - 484

$a^{2} + b^{2} - 484$ over the complex numbers.

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Step 2.1

Use the quadratic formula to find the roots for

a^{2} + b^{2} - 484 = 0

$a^{2} + b^{2} - 484 = 0$

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Step 2.1.1

Use the quadratic formula to find the solutions.

\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} = 0

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} = 0$

Step 2.1.2

Substitute the values

a = 1

$a = 1$ ,

b = 0

$b = 0$ , and

c = b^{2} - 484

$c = b^{2} - 484$ into the quadratic formula and solve for

a

$a$ .

\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot (b^{2} - 484))}}{2 \cdot 1} = 0

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot (b^{2} - 484))}}{2 \cdot 1} = 0$

Step 2.1.3

Simplify.

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Step 2.1.3.1

Simplify the numerator.

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Step 2.1.3.1.1

Raising

0

$0$ to any positive power yields

0

$0$ .

a = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot (b^{2} - 484)}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot (b^{2} - 484)}}{2 \cdot 1}$

Step 2.1.3.1.2

Multiply

- 4

$- 4$ by

1

$1$ .

a = \frac{0 \pm \sqrt{0 - 4 \cdot (b^{2} - 484)}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{0 - 4 \cdot (b^{2} - 484)}}{2 \cdot 1}$

Step 2.1.3.1.3

Apply the distributive property.

a = \frac{0 \pm \sqrt{0 - 4 b^{2} - 4 \cdot - 484}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{0 - 4 b^{2} - 4 \cdot - 484}}{2 \cdot 1}$

Step 2.1.3.1.4

Multiply

- 4

$- 4$ by

- 484

$- 484$ .

a = \frac{0 \pm \sqrt{0 - 4 b^{2} + 1936}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{0 - 4 b^{2} + 1936}}{2 \cdot 1}$

Step 2.1.3.1.5

Subtract

- (- 4 b^{2} + 1936)

$- (- 4 b^{2} + 1936)$ from

0

$0$ .

a = \frac{0 \pm \sqrt{- 4 b^{2} + 1936}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{- 4 b^{2} + 1936}}{2 \cdot 1}$

Step 2.1.3.1.6

Rewrite

- 4 b^{2} + 1936

$- 4 b^{2} + 1936$ in a factored form.

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Step 2.1.3.1.6.1

Factor

4

$4$ out of

- 4 b^{2} + 1936

$- 4 b^{2} + 1936$ .

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Step 2.1.3.1.6.1.1

Factor

4

$4$ out of

- 4 b^{2}

$- 4 b^{2}$ .

a = \frac{0 \pm \sqrt{4 (- b^{2}) + 1936}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{4 (- b^{2}) + 1936}}{2 \cdot 1}$

Step 2.1.3.1.6.1.2

Factor

4

$4$ out of

1936

$1936$ .

a = \frac{0 \pm \sqrt{4 (- b^{2}) + 4 (484)}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{4 (- b^{2}) + 4 (484)}}{2 \cdot 1}$

Step 2.1.3.1.6.1.3

Factor

4

$4$ out of

4 (- b^{2}) + 4 (484)

$4 (- b^{2}) + 4 (484)$ .

a = \frac{0 \pm \sqrt{4 (- b^{2} + 484)}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{4 (- b^{2} + 484)}}{2 \cdot 1}$

a = \frac{0 \pm \sqrt{4 (- b^{2} + 484)}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{4 (- b^{2} + 484)}}{2 \cdot 1}$

Step 2.1.3.1.6.2

Rewrite

484

$484$ as

22^{2}

$22^{2}$ .

a = \frac{0 \pm \sqrt{4 (- b^{2} + 22^{2})}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{4 (- b^{2} + 22^{2})}}{2 \cdot 1}$

Step 2.1.3.1.6.3

Reorder

- b^{2}

$- b^{2}$ and

22^{2}

$22^{2}$ .

a = \frac{0 \pm \sqrt{4 (22^{2} - b^{2})}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{4 (22^{2} - b^{2})}}{2 \cdot 1}$

Step 2.1.3.1.6.4

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 22

$a = 22$ and

b = b

$b = b$ .

a = \frac{0 \pm \sqrt{4 (22 + b) (22 - b)}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{4 (22 + b) (22 - b)}}{2 \cdot 1}$

a = \frac{0 \pm \sqrt{4 (22 + b) (22 - b)}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{4 (22 + b) (22 - b)}}{2 \cdot 1}$

Step 2.1.3.1.7

Rewrite

4 (22 + b) (22 - b)

$4 (22 + b) (22 - b)$ as

2^{2} ((22 + b) (22 - b))

$2^{2} ((22 + b) (22 - b))$ .

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Step 2.1.3.1.7.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

a = \frac{0 \pm \sqrt{2^{2} (22 + b) (22 - b)}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{2^{2} (22 + b) (22 - b)}}{2 \cdot 1}$

Step 2.1.3.1.7.2

Add parentheses.

a = \frac{0 \pm \sqrt{2^{2} ((22 + b) (22 - b))}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{2^{2} ((22 + b) (22 - b))}}{2 \cdot 1}$

a = \frac{0 \pm \sqrt{2^{2} ((22 + b) (22 - b))}}{2 \cdot 1}

$a = \frac{0 \pm \sqrt{2^{2} ((22 + b) (22 - b))}}{2 \cdot 1}$

Step 2.1.3.1.8

Pull terms out from under the radical.

a = \frac{0 \pm 2 \sqrt{(22 + b) (22 - b)}}{2 \cdot 1}

$a = \frac{0 \pm 2 \sqrt{(22 + b) (22 - b)}}{2 \cdot 1}$

a = \frac{0 \pm 2 \sqrt{(22 + b) (22 - b)}}{2 \cdot 1}

$a = \frac{0 \pm 2 \sqrt{(22 + b) (22 - b)}}{2 \cdot 1}$

Step 2.1.3.2

Multiply

2

$2$ by

1

$1$ .

a = \frac{0 \pm 2 \sqrt{(22 + b) (22 - b)}}{2}

$a = \frac{0 \pm 2 \sqrt{(22 + b) (22 - b)}}{2}$

Step 2.1.3.3

Simplify

\frac{0 \pm 2 \sqrt{(22 + b) (22 - b)}}{2}

$\frac{0 \pm 2 \sqrt{(22 + b) (22 - b)}}{2}$ .

a = \pm \sqrt{(22 + b) (22 - b)}

$a = \pm \sqrt{(22 + b) (22 - b)}$

Step 2.2

Find the factors from the roots, then multiply the factors together.

$(a - \sqrt{(22 + b) (22 - b)}) (a - (- \sqrt{(22 + b) (22 - b)})) = 0$

Step 2.3

Simplify the factored form.

$(a - \sqrt{(22 + b) (22 - b)}) (a + \sqrt{(22 + b) (22 - b)}) = 0$