Finite Math Examples

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

Step 1

Subtract $484$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Use the quadratic formula to find the roots for $a^{2} + b^{2} - 484 = 0$

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Step 2.1.1

Use the quadratic formula to find the solutions.

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

Step 2.1.2

Substitute the values $a = 1$ , $b = 0$ , and $c = b^{2} - 484$ into the quadratic formula and solve for $a$ .

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

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 $0$ to any positive power yields $0$ .

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

Step 2.1.3.1.2

Multiply $- 4$ by $1$ .

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

Step 2.1.3.1.3

Apply the distributive property.

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

Step 2.1.3.1.4

Multiply $- 4$ by $- 484$ .

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

Step 2.1.3.1.5

Subtract $- (- 4 b^{2} + 1936)$ from $0$ .

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

Step 2.1.3.1.6

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

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Step 2.1.3.1.6.1

Factor $4$ out of $- 4 b^{2} + 1936$ .

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Step 2.1.3.1.6.1.1

Factor $4$ out of $- 4 b^{2}$ .

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

Step 2.1.3.1.6.1.2

Factor $4$ out of $1936$ .

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

Step 2.1.3.1.6.1.3

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

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

Step 2.1.3.1.6.2

Rewrite $484$ as $22^{2}$ .

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

Step 2.1.3.1.6.3

Reorder $- b^{2}$ and $22^{2}$ .

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

Step 2.1.3.1.6.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 22$ and $b = b$ .

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

Step 2.1.3.1.7

Rewrite $4 (22 + b) (22 - b)$ as $2^{2} ((22 + b) (22 - b))$ .

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Step 2.1.3.1.7.1

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

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

Step 2.1.3.1.7.2

Add parentheses.

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

Step 2.1.3.1.8

Pull terms out from under the radical.

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

Step 2.1.3.2

Multiply $2$ by $1$ .

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

Step 2.1.3.3

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

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

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.

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