Algebra Examples

$A^{2} + B^{2}$

Use the quadratic formula to find the roots for $A^{2} + B^{2} = 0$

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Use the quadratic formula to find the solutions.

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

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

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

Simplify.

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Simplify the numerator.

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

$A = \frac{0 \pm \sqrt{0^{2} - {(2 \cdot (1 B))}^{2}}}{2 \cdot 1}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 0$ and $b = 2 \cdot 1 B$ .

$A = \frac{0 \pm \sqrt{(0 + 2 \cdot (1 B)) (0 - (2 \cdot (1 B)))}}{2 \cdot 1}$

Simplify.

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Multiply $2$ by $1$ .

$A = \frac{0 \pm \sqrt{(0 + 2 B) (0 - (2 \cdot (1 B)))}}{2 \cdot 1}$

Add $0$ and $2 B$ .

$A = \frac{0 \pm \sqrt{2 B (0 - (2 \cdot (1 B)))}}{2 \cdot 1}$

Combine exponents.

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Multiply $2$ by $1$ .

$A = \frac{0 \pm \sqrt{2 B (0 - (2 B))}}{2 \cdot 1}$

Multiply $2$ by $- 1$ .

$A = \frac{0 \pm \sqrt{2 B (0 - 2 B)}}{2 \cdot 1}$

Subtract $2 B$ from $0$ .

$A = \frac{0 \pm \sqrt{2 B \cdot (- 2 B)}}{2 \cdot 1}$

Combine exponents.

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Multiply $- 2$ by $2$ .

$A = \frac{0 \pm \sqrt{- 4 B \cdot B}}{2 \cdot 1}$

Raise $B$ to the power of $1$ .

$A = \frac{0 \pm \sqrt{- 4 (B \cdot B)}}{2 \cdot 1}$

Raise $B$ to the power of $1$ .

$A = \frac{0 \pm \sqrt{- 4 (B \cdot B)}}{2 \cdot 1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$A = \frac{0 \pm \sqrt{- 4 B^{1 + 1}}}{2 \cdot 1}$

Add $1$ and $1$ .

$A = \frac{0 \pm \sqrt{- 4 B^{2}}}{2 \cdot 1}$

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

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Factor $4$ out of $- 4$ .

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

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

$A = \frac{0 \pm \sqrt{2^{2} \cdot (- 1 B^{2})}}{2 \cdot 1}$

Move $- 1$ .

$A = \frac{0 \pm \sqrt{2^{2} B^{2} \cdot - 1}}{2 \cdot 1}$

Rewrite $2^{2} B^{2}$ as ${(2 B)}^{2}$ .

$A = \frac{0 \pm \sqrt{{(2 B)}^{2} \cdot - 1}}{2 \cdot 1}$

Pull terms out from under the radical.

$A = \frac{0 \pm 2 B \sqrt{- 1}}{2 \cdot 1}$

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

$A = \frac{0 \pm 2 B i}{2 \cdot 1}$

Multiply $2$ by $1$ .

$A = \frac{0 \pm 2 B i}{2}$

Simplify $\frac{0 \pm 2 B i}{2}$ .

$A = \pm B i$

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

$(A - (B i)) (A - (- B i))$

Simplify the factored form.

$(A - B i) (A + B i)$