Algebra Examples

$x^{2} + 4 y^{2} = 4$ $x^{2} + y^{2} = 4$

Step 1

Solve for $x$ in $x^{2} + 4 y^{2} = 4$ .

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Step 1.1

Subtract $4 y^{2}$ from both sides of the equation.

$x^{2} = 4 - 4 y^{2}$

$x^{2} + y^{2} = 4$

Step 1.2

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

$x = \pm \sqrt{4 - 4 y^{2}}$

$x^{2} + y^{2} = 4$

Step 1.3

Simplify $\pm \sqrt{4 - 4 y^{2}}$ .

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Step 1.3.1

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

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Step 1.3.1.1

Factor $4$ out of $4$ .

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

$x^{2} + y^{2} = 4$

Step 1.3.1.2

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

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

$x^{2} + y^{2} = 4$

Step 1.3.1.3

Factor $4$ out of $4 (1) + 4 (- y^{2})$ .

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

$x^{2} + y^{2} = 4$

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

$x^{2} + y^{2} = 4$

Step 1.3.2

Rewrite $1$ as $1^{2}$ .

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

$x^{2} + y^{2} = 4$

Step 1.3.3

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

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

$x^{2} + y^{2} = 4$

Step 1.3.4

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

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Step 1.3.4.1

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

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

$x^{2} + y^{2} = 4$

Step 1.3.4.2

Rewrite $1$ as $1^{2}$ .

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

$x^{2} + y^{2} = 4$

Step 1.3.4.3

Add parentheses.

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

$x^{2} + y^{2} = 4$

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

$x^{2} + y^{2} = 4$

Step 1.3.5

Pull terms out from under the radical.

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

$x^{2} + y^{2} = 4$

Step 1.3.6

One to any power is one.

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

$x^{2} + y^{2} = 4$

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

$x^{2} + y^{2} = 4$

Step 1.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.4.1

First, use the positive value of the $\pm$ to find the first solution.

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

$x^{2} + y^{2} = 4$

Step 1.4.2

Next, use the negative value of the $\pm$ to find the second solution.

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

$x^{2} + y^{2} = 4$

Step 1.4.3

The complete solution is the result of both the positive and negative portions of the solution.

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

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

$x^{2} + y^{2} = 4$

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

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

$x^{2} + y^{2} = 4$

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

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

$x^{2} + y^{2} = 4$

Step 2

Solve the system $x = 2 \sqrt{(1 + y) (1 - y)}, x^{2} + y^{2} = 4$ .

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Step 2.1

Replace all occurrences of $x$ with $2 \sqrt{(1 + y) (1 - y)}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 4$ with $2 \sqrt{(1 + y) (1 - y)}$ .

${(2 \sqrt{(1 + y) (1 - y)})}^{2} + y^{2} = 4$

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

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

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

Apply the product rule to $2 \sqrt{(1 + y) (1 - y)}$ .

$2^{2} {\sqrt{(1 + y) (1 - y)}}^{2} + y^{2} = 4$

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

Step 2.1.2.1.1.2

Raise $2$ to the power of $2$ .

$4 {\sqrt{(1 + y) (1 - y)}}^{2} + y^{2} = 4$

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

Step 2.1.2.1.1.3

Rewrite ${\sqrt{(1 + y) (1 - y)}}^{2}$ as $(1 + y) (1 - y)$ .

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Step 2.1.2.1.1.3.1

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

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

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

Step 2.1.2.1.1.3.2

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

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

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

Step 2.1.2.1.1.3.3

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

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

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

Step 2.1.2.1.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.3.4.1

Cancel the common factor.

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

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

Step 2.1.2.1.1.3.4.2

Rewrite the expression.

$4 ((1 + y) (1 - y)) + y^{2} = 4$

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

$4 ((1 + y) (1 - y)) + y^{2} = 4$

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

Step 2.1.2.1.1.3.5

Simplify.

$4 ((1 + y) (1 - y)) + y^{2} = 4$

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

$4 ((1 + y) (1 - y)) + y^{2} = 4$

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

Step 2.1.2.1.1.4

Expand $(1 + y) (1 - y)$ using the FOIL Method.

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Step 2.1.2.1.1.4.1

Apply the distributive property.

$4 (1 (1 - y) + y (1 - y)) + y^{2} = 4$

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

Step 2.1.2.1.1.4.2

Apply the distributive property.

$4 (1 \cdot 1 + 1 (- y) + y (1 - y)) + y^{2} = 4$

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

Step 2.1.2.1.1.4.3

Apply the distributive property.

$4 (1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y)) + y^{2} = 4$

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

$4 (1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y)) + y^{2} = 4$

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

Step 2.1.2.1.1.5

Simplify and combine like terms.

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Step 2.1.2.1.1.5.1

Simplify each term.

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Step 2.1.2.1.1.5.1.1

Multiply $1$ by $1$ .

$4 (1 + 1 (- y) + y \cdot 1 + y (- y)) + y^{2} = 4$

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

Step 2.1.2.1.1.5.1.2

Multiply $- y$ by $1$ .

$4 (1 - y + y \cdot 1 + y (- y)) + y^{2} = 4$

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

Step 2.1.2.1.1.5.1.3

Multiply $y$ by $1$ .

$4 (1 - y + y + y (- y)) + y^{2} = 4$

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

Step 2.1.2.1.1.5.1.4

Rewrite using the commutative property of multiplication.

$4 (1 - y + y - y \cdot y) + y^{2} = 4$

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

Step 2.1.2.1.1.5.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.2.1.1.5.1.5.1

Move $y$ .

$4 (1 - y + y - (y \cdot y)) + y^{2} = 4$

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

Step 2.1.2.1.1.5.1.5.2

Multiply $y$ by $y$ .

$4 (1 - y + y - y^{2}) + y^{2} = 4$

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

$4 (1 - y + y - y^{2}) + y^{2} = 4$

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

$4 (1 - y + y - y^{2}) + y^{2} = 4$

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

Step 2.1.2.1.1.5.2

Add $- y$ and $y$ .

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

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

Step 2.1.2.1.1.5.3

Add $1$ and $0$ .

$4 (1 - y^{2}) + y^{2} = 4$

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

$4 (1 - y^{2}) + y^{2} = 4$

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

Step 2.1.2.1.1.6

Apply the distributive property.

$4 \cdot 1 + 4 (- y^{2}) + y^{2} = 4$

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

Step 2.1.2.1.1.7

Multiply $4$ by $1$ .

$4 + 4 (- y^{2}) + y^{2} = 4$

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

Step 2.1.2.1.1.8

Multiply $- 1$ by $4$ .

$4 - 4 y^{2} + y^{2} = 4$

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

$4 - 4 y^{2} + y^{2} = 4$

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

Step 2.1.2.1.2

Add $- 4 y^{2}$ and $y^{2}$ .

$4 - 3 y^{2} = 4$

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

$4 - 3 y^{2} = 4$

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

$4 - 3 y^{2} = 4$

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

$4 - 3 y^{2} = 4$

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

Step 2.2

Solve for $y$ in $4 - 3 y^{2} = 4$ .

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Step 2.2.1

Move all terms not containing $y$ to the right side of the equation.

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Step 2.2.1.1

Subtract $4$ from both sides of the equation.

$- 3 y^{2} = 4 - 4$

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

Step 2.2.1.2

Subtract $4$ from $4$ .

$- 3 y^{2} = 0$

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

$- 3 y^{2} = 0$

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

Step 2.2.2

Divide each term in $- 3 y^{2} = 0$ by $- 3$ and simplify.

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Step 2.2.2.1

Divide each term in $- 3 y^{2} = 0$ by $- 3$ .

$\frac{- 3 y^{2}}{- 3} = \frac{0}{- 3}$

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

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 y^{2}}{- 3} = \frac{0}{- 3}$

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

Step 2.2.2.2.1.2

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

$y^{2} = \frac{0}{- 3}$

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

$y^{2} = \frac{0}{- 3}$

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

$y^{2} = \frac{0}{- 3}$

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

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $0$ by $- 3$ .

$y^{2} = 0$

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

$y^{2} = 0$

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

$y^{2} = 0$

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

Step 2.2.3

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

$y = \pm \sqrt{0}$

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

Step 2.2.4

Simplify $\pm \sqrt{0}$ .

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Step 2.2.4.1

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

$y = \pm \sqrt{0^{2}}$

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

Step 2.2.4.2

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

$y = \pm 0$

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

Step 2.2.4.3

Plus or minus $0$ is $0$ .

$y = 0$

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

$y = 0$

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

$y = 0$

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

Step 2.3

Replace all occurrences of $y$ with $0$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $x = 2 \sqrt{(1 + y) (1 - y)}$ with $0$ .

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

$y = 0$

Step 2.3.2

Simplify $x = 2 \sqrt{(1 + 0) (1 - (0))}$ .

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Step 2.3.2.1

Simplify the left side.

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Step 2.3.2.1.1

Remove parentheses.

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

$y = 0$

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

$y = 0$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Simplify $2 \sqrt{(1 + 0) (1 - (0))}$ .

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Step 2.3.2.2.1.1

Add $1$ and $0$ .

$x = 2 \sqrt{1 (1 - (0))}$

$y = 0$

Step 2.3.2.2.1.2

Multiply $1 - (0)$ by $1$ .

$x = 2 \sqrt{1 - (0)}$

$y = 0$

Step 2.3.2.2.1.3

Multiply $- 1$ by $0$ .

$x = 2 \sqrt{1 + 0}$

$y = 0$

Step 2.3.2.2.1.4

Add $1$ and $0$ .

$x = 2 \sqrt{1}$

$y = 0$

Step 2.3.2.2.1.5

Any root of $1$ is $1$ .

$x = 2 \cdot 1$

$y = 0$

Step 2.3.2.2.1.6

Multiply $2$ by $1$ .

$x = 2$

$y = 0$

$x = 2$

$y = 0$

$x = 2$

$y = 0$

$x = 2$

$y = 0$

$x = 2$

$y = 0$

$x = 2$

$y = 0$

Step 3

Solve the system $x = - 2 \sqrt{(1 + y) (1 - y)}, x^{2} + y^{2} = 4$ .

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Step 3.1

Replace all occurrences of $x$ with $- 2 \sqrt{(1 + y) (1 - y)}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 4$ with $- 2 \sqrt{(1 + y) (1 - y)}$ .

${(- 2 \sqrt{(1 + y) (1 - y)})}^{2} + y^{2} = 4$

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

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

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Apply the product rule to $- 2 \sqrt{(1 + y) (1 - y)}$ .

${(- 2)}^{2} {\sqrt{(1 + y) (1 - y)}}^{2} + y^{2} = 4$

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

Step 3.1.2.1.1.2

Raise $- 2$ to the power of $2$ .

$4 {\sqrt{(1 + y) (1 - y)}}^{2} + y^{2} = 4$

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

Step 3.1.2.1.1.3

Rewrite ${\sqrt{(1 + y) (1 - y)}}^{2}$ as $(1 + y) (1 - y)$ .

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Step 3.1.2.1.1.3.1

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

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

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

Step 3.1.2.1.1.3.2

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

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

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

Step 3.1.2.1.1.3.3

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

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

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

Step 3.1.2.1.1.3.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.3.4.1

Cancel the common factor.

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

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

Step 3.1.2.1.1.3.4.2

Rewrite the expression.

$4 ((1 + y) (1 - y)) + y^{2} = 4$

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

$4 ((1 + y) (1 - y)) + y^{2} = 4$

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

Step 3.1.2.1.1.3.5

Simplify.

$4 ((1 + y) (1 - y)) + y^{2} = 4$

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

$4 ((1 + y) (1 - y)) + y^{2} = 4$

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

Step 3.1.2.1.1.4

Expand $(1 + y) (1 - y)$ using the FOIL Method.

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Step 3.1.2.1.1.4.1

Apply the distributive property.

$4 (1 (1 - y) + y (1 - y)) + y^{2} = 4$

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

Step 3.1.2.1.1.4.2

Apply the distributive property.

$4 (1 \cdot 1 + 1 (- y) + y (1 - y)) + y^{2} = 4$

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

Step 3.1.2.1.1.4.3

Apply the distributive property.

$4 (1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y)) + y^{2} = 4$

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

$4 (1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y)) + y^{2} = 4$

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

Step 3.1.2.1.1.5

Simplify and combine like terms.

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Step 3.1.2.1.1.5.1

Simplify each term.

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Step 3.1.2.1.1.5.1.1

Multiply $1$ by $1$ .

$4 (1 + 1 (- y) + y \cdot 1 + y (- y)) + y^{2} = 4$

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

Step 3.1.2.1.1.5.1.2

Multiply $- y$ by $1$ .

$4 (1 - y + y \cdot 1 + y (- y)) + y^{2} = 4$

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

Step 3.1.2.1.1.5.1.3

Multiply $y$ by $1$ .

$4 (1 - y + y + y (- y)) + y^{2} = 4$

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

Step 3.1.2.1.1.5.1.4

Rewrite using the commutative property of multiplication.

$4 (1 - y + y - y \cdot y) + y^{2} = 4$

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

Step 3.1.2.1.1.5.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.2.1.1.5.1.5.1

Move $y$ .

$4 (1 - y + y - (y \cdot y)) + y^{2} = 4$

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

Step 3.1.2.1.1.5.1.5.2

Multiply $y$ by $y$ .

$4 (1 - y + y - y^{2}) + y^{2} = 4$

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

$4 (1 - y + y - y^{2}) + y^{2} = 4$

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

$4 (1 - y + y - y^{2}) + y^{2} = 4$

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

Step 3.1.2.1.1.5.2

Add $- y$ and $y$ .

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

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

Step 3.1.2.1.1.5.3

Add $1$ and $0$ .

$4 (1 - y^{2}) + y^{2} = 4$

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

$4 (1 - y^{2}) + y^{2} = 4$

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

Step 3.1.2.1.1.6

Apply the distributive property.

$4 \cdot 1 + 4 (- y^{2}) + y^{2} = 4$

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

Step 3.1.2.1.1.7

Multiply $4$ by $1$ .

$4 + 4 (- y^{2}) + y^{2} = 4$

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

Step 3.1.2.1.1.8

Multiply $- 1$ by $4$ .

$4 - 4 y^{2} + y^{2} = 4$

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

$4 - 4 y^{2} + y^{2} = 4$

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

Step 3.1.2.1.2

Add $- 4 y^{2}$ and $y^{2}$ .

$4 - 3 y^{2} = 4$

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

$4 - 3 y^{2} = 4$

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

$4 - 3 y^{2} = 4$

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

$4 - 3 y^{2} = 4$

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

Step 3.2

Solve for $y$ in $4 - 3 y^{2} = 4$ .

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Step 3.2.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.2.1.1

Subtract $4$ from both sides of the equation.

$- 3 y^{2} = 4 - 4$

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

Step 3.2.1.2

Subtract $4$ from $4$ .

$- 3 y^{2} = 0$

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

$- 3 y^{2} = 0$

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

Step 3.2.2

Divide each term in $- 3 y^{2} = 0$ by $- 3$ and simplify.

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Step 3.2.2.1

Divide each term in $- 3 y^{2} = 0$ by $- 3$ .

$\frac{- 3 y^{2}}{- 3} = \frac{0}{- 3}$

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 y^{2}}{- 3} = \frac{0}{- 3}$

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

Step 3.2.2.2.1.2

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

$y^{2} = \frac{0}{- 3}$

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

$y^{2} = \frac{0}{- 3}$

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

$y^{2} = \frac{0}{- 3}$

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

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $0$ by $- 3$ .

$y^{2} = 0$

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

$y^{2} = 0$

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

$y^{2} = 0$

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

Step 3.2.3

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

$y = \pm \sqrt{0}$

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

Step 3.2.4

Simplify $\pm \sqrt{0}$ .

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Step 3.2.4.1

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

$y = \pm \sqrt{0^{2}}$

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

Step 3.2.4.2

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

$y = \pm 0$

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

Step 3.2.4.3

Plus or minus $0$ is $0$ .

$y = 0$

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

$y = 0$

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

$y = 0$

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

Step 3.3

Replace all occurrences of $y$ with $0$ in each equation.

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Step 3.3.1

Replace all occurrences of $y$ in $x = - 2 \sqrt{(1 + y) (1 - y)}$ with $0$ .

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

$y = 0$

Step 3.3.2

Simplify $x = - 2 \sqrt{(1 + 0) (1 - (0))}$ .

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Step 3.3.2.1

Simplify the left side.

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Step 3.3.2.1.1

Remove parentheses.

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

$y = 0$

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

$y = 0$

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Simplify $- 2 \sqrt{(1 + 0) (1 - (0))}$ .

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Step 3.3.2.2.1.1

Add $1$ and $0$ .

$x = - 2 \sqrt{1 (1 - (0))}$

$y = 0$

Step 3.3.2.2.1.2

Multiply $1 - (0)$ by $1$ .

$x = - 2 \sqrt{1 - (0)}$

$y = 0$

Step 3.3.2.2.1.3

Multiply $- 1$ by $0$ .

$x = - 2 \sqrt{1 + 0}$

$y = 0$

Step 3.3.2.2.1.4

Add $1$ and $0$ .

$x = - 2 \sqrt{1}$

$y = 0$

Step 3.3.2.2.1.5

Any root of $1$ is $1$ .

$x = - 2 \cdot 1$

$y = 0$

Step 3.3.2.2.1.6

Multiply $- 2$ by $1$ .

$x = - 2$

$y = 0$

$x = - 2$

$y = 0$

$x = - 2$

$y = 0$

$x = - 2$

$y = 0$

$x = - 2$

$y = 0$

$x = - 2$

$y = 0$

Step 4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(2, 0)$

$(- 2, 0)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(2, 0), (- 2, 0)$

Equation Form:

$x = 2, y = 0$

$x = - 2, y = 0$

Step 6