Algebra Examples

$r^{2} - 2 r \sin (θ) = 0$

Step 1

Since $\sin (θ) = \frac{y}{r}$ , replace $\sin (θ)$ with $\frac{y}{r}$ .

$r^{2} - 2 r \frac{y}{r} = 0$

Step 2

Since $r^{2} = x^{2} + y^{2}$ , replace $r^{2}$ with $x^{2} + y^{2}$ and $r$ with $\sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} - 2 \sqrt{x^{2} + y^{2}} \frac{y}{\sqrt{x^{2} + y^{2}}} = 0$

Step 3

Write in standard form.

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Step 3.1

Solve for $y$ .

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Step 3.1.1

To remove the radical on the left side of the equation, square both sides of the equation.

${(x^{2} + y^{2} - 2 \sqrt{x^{2} + y^{2}} \frac{y}{\sqrt{x^{2} + y^{2}}})}^{2} = 0^{2}$

Step 3.1.2

Simplify each side of the equation.

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Step 3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + y^{2}}$ as ${(x^{2} + y^{2})}^{\frac{1}{2}}$ .

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

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Simplify ${(x^{2} + y^{2} - 2 {(x^{2} + y^{2})}^{\frac{1}{2}} \frac{y}{\sqrt{x^{2} + y^{2}}})}^{2}$ .

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Step 3.1.2.2.1.1

Simplify each term.

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Step 3.1.2.2.1.1.1

Multiply $\frac{y}{\sqrt{x^{2} + y^{2}}}$ by $\frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}}$ .

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

Step 3.1.2.2.1.1.2

Combine and simplify the denominator.

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Step 3.1.2.2.1.1.2.1

Multiply $\frac{y}{\sqrt{x^{2} + y^{2}}}$ by $\frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}}$ .

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

Step 3.1.2.2.1.1.2.2

Raise $\sqrt{x^{2} + y^{2}}$ to the power of $1$ .

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

Step 3.1.2.2.1.1.2.3

Raise $\sqrt{x^{2} + y^{2}}$ to the power of $1$ .

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

Step 3.1.2.2.1.1.2.4

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

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

Step 3.1.2.2.1.1.2.5

Add $1$ and $1$ .

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

Step 3.1.2.2.1.1.2.6

Rewrite ${\sqrt{x^{2} + y^{2}}}^{2}$ as $x^{2} + y^{2}$ .

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Step 3.1.2.2.1.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + y^{2}}$ as ${(x^{2} + y^{2})}^{\frac{1}{2}}$ .

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

Step 3.1.2.2.1.1.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.1.2.2.1.1.2.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.1.2.2.1.1.2.6.4

Cancel the common factor of $2$ .

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Step 3.1.2.2.1.1.2.6.4.1

Cancel the common factor.

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

Step 3.1.2.2.1.1.2.6.4.2

Rewrite the expression.

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

Step 3.1.2.2.1.1.2.6.5

Simplify.

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

Step 3.1.2.2.1.1.3

Multiply $- 2 {(x^{2} + y^{2})}^{\frac{1}{2}} \frac{y \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$ .

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Step 3.1.2.2.1.1.3.1

Combine $\frac{y \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2}}$ and $- 2$ .

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

Step 3.1.2.2.1.1.3.2

Combine $\frac{y \sqrt{x^{2} + y^{2}} \cdot - 2}{x^{2} + y^{2}}$ and ${(x^{2} + y^{2})}^{\frac{1}{2}}$ .

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

Step 3.1.2.2.1.1.3.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + y^{2}}$ as ${(x^{2} + y^{2})}^{\frac{1}{2}}$ .

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

Step 3.1.2.2.1.1.3.4

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

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

Step 3.1.2.2.1.1.3.5

Combine the numerators over the common denominator.

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

Step 3.1.2.2.1.1.3.6

Add $1$ and $1$ .

${(x^{2} + y^{2} + \frac{y \cdot - 2 {(x^{2} + y^{2})}^{\frac{2}{2}}}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.1.3.7

Cancel the common factor of $2$ .

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Step 3.1.2.2.1.1.3.7.1

Cancel the common factor.

${(x^{2} + y^{2} + \frac{y \cdot - 2 {(x^{2} + y^{2})}^{\frac{2}{2}}}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.1.3.7.2

Rewrite the expression.

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

Step 3.1.2.2.1.1.4

Move $- 2$ to the left of $y$ .

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

Step 3.1.2.2.1.1.5

Move the negative in front of the fraction.

${(x^{2} + y^{2} - \frac{2 y {(x^{2} + y^{2})}^{1}}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.2

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + y^{2}}{x^{2} + y^{2}}$ .

${(y^{2} + \frac{x^{2} (x^{2} + y^{2})}{x^{2} + y^{2}} - \frac{2 y {(x^{2} + y^{2})}^{1}}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.3

Combine the numerators over the common denominator.

${(y^{2} + \frac{x^{2} (x^{2} + y^{2}) - 2 y {(x^{2} + y^{2})}^{1}}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.4

Simplify each term.

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Step 3.1.2.2.1.4.1

Simplify the numerator.

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Step 3.1.2.2.1.4.1.1

Apply the distributive property.

${(y^{2} + \frac{x^{2} x^{2} + x^{2} y^{2} - 2 y {(x^{2} + y^{2})}^{1}}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.4.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.1.2.2.1.4.1.2.1

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

${(y^{2} + \frac{x^{2 + 2} + x^{2} y^{2} - 2 y {(x^{2} + y^{2})}^{1}}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.4.1.2.2

Add $2$ and $2$ .

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

Step 3.1.2.2.1.4.1.3

Simplify.

${(y^{2} + \frac{x^{4} + x^{2} y^{2} - 2 y (x^{2} + y^{2})}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.4.1.4

Apply the distributive property.

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

Step 3.1.2.2.1.4.1.5

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 3.1.2.2.1.4.1.5.1

Move $y^{2}$ .

${(y^{2} + \frac{x^{4} + x^{2} y^{2} - 2 y x^{2} - 2 (y^{2} y)}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.4.1.5.2

Multiply $y^{2}$ by $y$ .

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Step 3.1.2.2.1.4.1.5.2.1

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

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

Step 3.1.2.2.1.4.1.5.2.2

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

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

Step 3.1.2.2.1.4.1.5.3

Add $2$ and $1$ .

${(y^{2} + \frac{x^{4} + x^{2} y^{2} - 2 y x^{2} - 2 y^{3}}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.4.1.6

Rewrite $x^{4} + x^{2} y^{2} - 2 y x^{2} - 2 y^{3}$ in a factored form.

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Step 3.1.2.2.1.4.1.6.1

Factor out the greatest common factor from each group.

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Step 3.1.2.2.1.4.1.6.1.1

Group the first two terms and the last two terms.

${(y^{2} + \frac{(x^{4} + x^{2} y^{2}) - 2 y x^{2} - 2 y^{3}}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.4.1.6.1.2

Factor out the greatest common factor (GCF) from each group.

${(y^{2} + \frac{x^{2} (x^{2} + y^{2}) - 2 y (x^{2} + y^{2})}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.4.1.6.2

Factor the polynomial by factoring out the greatest common factor, $x^{2} + y^{2}$ .

${(y^{2} + \frac{(x^{2} + y^{2}) (x^{2} - 2 y)}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.4.2

Cancel the common factor of $x^{2} + y^{2}$ .

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Step 3.1.2.2.1.4.2.1

Cancel the common factor.

${(y^{2} + \frac{(x^{2} + y^{2}) (x^{2} - 2 y)}{x^{2} + y^{2}})}^{2} = 0^{2}$

Step 3.1.2.2.1.4.2.2

Divide $x^{2} - 2 y$ by $1$ .

${(y^{2} + x^{2} - 2 y)}^{2} = 0^{2}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

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

${(y^{2} + x^{2} - 2 y)}^{2} = 0$

Step 3.1.3

Solve for $y$ .

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Step 3.1.3.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y^{2} + x^{2} - 2 y = \pm \sqrt{0}$

Step 3.1.3.2

Simplify $\pm \sqrt{0}$ .

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Step 3.1.3.2.1

Rewrite $0$ as $0^{2}$ .

$y^{2} + x^{2} - 2 y = \pm \sqrt{0^{2}}$

Step 3.1.3.2.2

Pull terms out from under the radical, assuming positive real numbers.

$y^{2} + x^{2} - 2 y = \pm 0$

Step 3.1.3.2.3

Plus or minus $0$ is $0$ .

$y^{2} + x^{2} - 2 y = 0$

Step 3.1.3.3

Use the quadratic formula to find the solutions.

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

Step 3.1.3.4

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

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

Step 3.1.3.5

Simplify.

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Step 3.1.3.5.1

Simplify the numerator.

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Step 3.1.3.5.1.1

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

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

Step 3.1.3.5.1.2

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

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

Step 3.1.3.5.1.3

Simplify.

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Step 3.1.3.5.1.3.1

Multiply $2$ by $1$ .

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

Step 3.1.3.5.1.3.2

Factor $2$ out of $- 2 + 2 x$ .

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Step 3.1.3.5.1.3.2.1

Factor $2$ out of $- 2$ .

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

Step 3.1.3.5.1.3.2.2

Factor $2$ out of $2 \cdot - 1 + 2 x$ .

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

Step 3.1.3.5.1.3.3

Combine exponents.

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Step 3.1.3.5.1.3.3.1

Multiply $2$ by $1$ .

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

Step 3.1.3.5.1.3.3.2

Multiply $2$ by $- 1$ .

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

Step 3.1.3.5.1.4

Factor $2$ out of $- 2 - 2 x$ .

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Step 3.1.3.5.1.4.1

Factor $2$ out of $- 2$ .

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

Step 3.1.3.5.1.4.2

Factor $2$ out of $- 2 x$ .

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

Step 3.1.3.5.1.4.3

Factor $2$ out of $2 (- 1) + 2 (- x)$ .

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

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

Step 3.1.3.5.1.5

Multiply $2$ by $2$ .

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

Step 3.1.3.5.1.6

Rewrite $4 (- 1 + x) (- 1 - x)$ as $2^{2} ((- 1 + x) (- 1 - x))$ .

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Step 3.1.3.5.1.6.1

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

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

Step 3.1.3.5.1.6.2

Add parentheses.

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

Step 3.1.3.5.1.7

Pull terms out from under the radical.

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

Step 3.1.3.5.2

Multiply $2$ by $1$ .

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

Step 3.1.3.5.3

Simplify $\frac{2 \pm 2 \sqrt{(- 1 + x) (- 1 - x)}}{2}$ .

$y = 1 \pm \sqrt{(- 1 + x) (- 1 - x)}$

Step 3.1.3.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.1.3.6.1

Simplify the numerator.

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Step 3.1.3.6.1.1

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

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

Step 3.1.3.6.1.2

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

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

Step 3.1.3.6.1.3

Simplify.

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Step 3.1.3.6.1.3.1

Multiply $2$ by $1$ .

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

Step 3.1.3.6.1.3.2

Factor $2$ out of $- 2 + 2 x$ .

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Step 3.1.3.6.1.3.2.1

Factor $2$ out of $- 2$ .

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

Step 3.1.3.6.1.3.2.2

Factor $2$ out of $2 \cdot - 1 + 2 x$ .

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

Step 3.1.3.6.1.3.3

Combine exponents.

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Step 3.1.3.6.1.3.3.1

Multiply $2$ by $1$ .

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

Step 3.1.3.6.1.3.3.2

Multiply $2$ by $- 1$ .

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

Step 3.1.3.6.1.4

Factor $2$ out of $- 2 - 2 x$ .

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Step 3.1.3.6.1.4.1

Factor $2$ out of $- 2$ .

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

Step 3.1.3.6.1.4.2

Factor $2$ out of $- 2 x$ .

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

Step 3.1.3.6.1.4.3

Factor $2$ out of $2 (- 1) + 2 (- x)$ .

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

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

Step 3.1.3.6.1.5

Multiply $2$ by $2$ .

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

Step 3.1.3.6.1.6

Rewrite $4 (- 1 + x) (- 1 - x)$ as $2^{2} ((- 1 + x) (- 1 - x))$ .

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Step 3.1.3.6.1.6.1

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

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

Step 3.1.3.6.1.6.2

Add parentheses.

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

Step 3.1.3.6.1.7

Pull terms out from under the radical.

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

Step 3.1.3.6.2

Multiply $2$ by $1$ .

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

Step 3.1.3.6.3

Simplify $\frac{2 \pm 2 \sqrt{(- 1 + x) (- 1 - x)}}{2}$ .

$y = 1 \pm \sqrt{(- 1 + x) (- 1 - x)}$

Step 3.1.3.6.4

Change the $\pm$ to $+$ .

$y = 1 + \sqrt{(- 1 + x) (- 1 - x)}$

Step 3.1.3.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.1.3.7.1

Simplify the numerator.

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Step 3.1.3.7.1.1

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

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

Step 3.1.3.7.1.2

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

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

Step 3.1.3.7.1.3

Simplify.

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Step 3.1.3.7.1.3.1

Multiply $2$ by $1$ .

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

Step 3.1.3.7.1.3.2

Factor $2$ out of $- 2 + 2 x$ .

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Step 3.1.3.7.1.3.2.1

Factor $2$ out of $- 2$ .

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

Step 3.1.3.7.1.3.2.2

Factor $2$ out of $2 \cdot - 1 + 2 x$ .

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

Step 3.1.3.7.1.3.3

Combine exponents.

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Step 3.1.3.7.1.3.3.1

Multiply $2$ by $1$ .

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

Step 3.1.3.7.1.3.3.2

Multiply $2$ by $- 1$ .

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

Step 3.1.3.7.1.4

Factor $2$ out of $- 2 - 2 x$ .

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Step 3.1.3.7.1.4.1

Factor $2$ out of $- 2$ .

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

Step 3.1.3.7.1.4.2

Factor $2$ out of $- 2 x$ .

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

Step 3.1.3.7.1.4.3

Factor $2$ out of $2 (- 1) + 2 (- x)$ .

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

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

Step 3.1.3.7.1.5

Multiply $2$ by $2$ .

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

Step 3.1.3.7.1.6

Rewrite $4 (- 1 + x) (- 1 - x)$ as $2^{2} ((- 1 + x) (- 1 - x))$ .

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Step 3.1.3.7.1.6.1

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

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

Step 3.1.3.7.1.6.2

Add parentheses.

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

Step 3.1.3.7.1.7

Pull terms out from under the radical.

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

Step 3.1.3.7.2

Multiply $2$ by $1$ .

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

Step 3.1.3.7.3

Simplify $\frac{2 \pm 2 \sqrt{(- 1 + x) (- 1 - x)}}{2}$ .

$y = 1 \pm \sqrt{(- 1 + x) (- 1 - x)}$

Step 3.1.3.7.4

Change the $\pm$ to $-$ .

$y = 1 - \sqrt{(- 1 + x) (- 1 - x)}$

Step 3.1.3.8

The final answer is the combination of both solutions.

$y = 1 + \sqrt{(- 1 + x) (- 1 - x)}$

$y = 1 - \sqrt{(- 1 + x) (- 1 - x)}$

$y = 1 + \sqrt{(- 1 + x) (- 1 - x)}$

$y = 1 - \sqrt{(- 1 + x) (- 1 - x)}$

$y = 1 + \sqrt{(- 1 + x) (- 1 - x)}$

$y = 1 - \sqrt{(- 1 + x) (- 1 - x)}$

Step 3.2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3.3

The standard form is $1 + \sqrt{(- 1 + x) (- 1 - x)}, 1 - \sqrt{(- 1 + x) (- 1 - x)}$ .

$y = 1 + \sqrt{(- 1 + x) (- 1 - x)}$

$y = 1 - \sqrt{(- 1 + x) (- 1 - x)}$

$y = 1 + \sqrt{(- 1 + x) (- 1 - x)}$

$y = 1 - \sqrt{(- 1 + x) (- 1 - x)}$

Step 4