Calculus Examples

$x^{2} + y^{2}$

Step 1

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

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Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Simplify.

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Step 1.3.1.3.1

Multiply $2$ by $1$ .

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

Step 1.3.1.3.2

Add $0$ and $2 y$ .

$x = \frac{0 \pm \sqrt{2 y (0 - (2 \cdot (1 y)))}}{2 \cdot 1}$

Step 1.3.1.3.3

Combine exponents.

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Step 1.3.1.3.3.1

Multiply $2$ by $1$ .

$x = \frac{0 \pm \sqrt{2 y (0 - (2 y))}}{2 \cdot 1}$

Step 1.3.1.3.3.2

Multiply $2$ by $- 1$ .

$x = \frac{0 \pm \sqrt{2 y (0 - 2 y)}}{2 \cdot 1}$

Step 1.3.1.4

Subtract $2 y$ from $0$ .

$x = \frac{0 \pm \sqrt{2 y \cdot (- 2 y)}}{2 \cdot 1}$

Step 1.3.1.5

Combine exponents.

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Step 1.3.1.5.1

Multiply $- 2$ by $2$ .

$x = \frac{0 \pm \sqrt{- 4 y \cdot y}}{2 \cdot 1}$

Step 1.3.1.5.2

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

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

Step 1.3.1.5.3

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

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

Step 1.3.1.5.4

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

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

Step 1.3.1.5.5

Add $1$ and $1$ .

$x = \frac{0 \pm \sqrt{- 4 y^{2}}}{2 \cdot 1}$

Step 1.3.1.6

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

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Step 1.3.1.6.1

Factor $4$ out of $- 4$ .

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

Step 1.3.1.6.2

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

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

Step 1.3.1.6.3

Move $- 1$ .

$x = \frac{0 \pm \sqrt{2^{2} y^{2} \cdot - 1}}{2 \cdot 1}$

Step 1.3.1.6.4

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

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

Step 1.3.1.7

Pull terms out from under the radical.

$x = \frac{0 \pm 2 y \sqrt{- 1}}{2 \cdot 1}$

Step 1.3.1.8

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

$x = \frac{0 \pm 2 y i}{2 \cdot 1}$

Step 1.3.2

Multiply $2$ by $1$ .

$x = \frac{0 \pm 2 y i}{2}$

Step 1.3.3

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

$x = \pm y i$

Step 2

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

$(x - (y i)) (x - (- y i))$

Step 3

Simplify the factored form.

$(x - y i) (x + y i)$