Algebra Examples

$x^{2} = 2 y^{2} + 4$ $x^{2} + 2 y^{2} = 68$

Step 1

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

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Step 1.1

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

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

$x^{2} + 2 y^{2} = 68$

Step 1.2

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

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Step 1.2.1

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

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

$x^{2} + 2 y^{2} = 68$

Step 1.2.2

Factor $2$ out of $4$ .

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

$x^{2} + 2 y^{2} = 68$

Step 1.2.3

Factor $2$ out of $2 y^{2} + 2 \cdot 2$ .

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

$x^{2} + 2 y^{2} = 68$

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

$x^{2} + 2 y^{2} = 68$

Step 1.3

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

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Step 1.3.1

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

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

$x^{2} + 2 y^{2} = 68$

Step 1.3.2

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

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

$x^{2} + 2 y^{2} = 68$

Step 1.3.3

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

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

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

$x^{2} + 2 y^{2} = 68$

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

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

$x^{2} + 2 y^{2} = 68$

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

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

$x^{2} + 2 y^{2} = 68$

Step 2

Solve the system $x = \sqrt{2 (y^{2} + 2)}, x^{2} + 2 y^{2} = 68$ .

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Step 2.1

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

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Step 2.1.1

Replace all occurrences of $x$ in $x^{2} + 2 y^{2} = 68$ with $\sqrt{2 (y^{2} + 2)}$ .

${(\sqrt{2 (y^{2} + 2)})}^{2} + 2 y^{2} = 68$

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

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

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

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

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Step 2.1.2.1.1.1.1

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

${({(2 (y^{2} + 2))}^{\frac{1}{2}})}^{2} + 2 y^{2} = 68$

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

Step 2.1.2.1.1.1.2

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

${(2 (y^{2} + 2))}^{\frac{1}{2} \cdot 2} + 2 y^{2} = 68$

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

Step 2.1.2.1.1.1.3

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

${(2 (y^{2} + 2))}^{\frac{2}{2}} + 2 y^{2} = 68$

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

Step 2.1.2.1.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.1.4.1

Cancel the common factor.

${(2 (y^{2} + 2))}^{\frac{2}{2}} + 2 y^{2} = 68$

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

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$(2 (y^{2} + 2)) + 2 y^{2} = 68$

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

$(2 (y^{2} + 2)) + 2 y^{2} = 68$

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

Step 2.1.2.1.1.1.5

Simplify.

$2 (y^{2} + 2) + 2 y^{2} = 68$

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

$2 (y^{2} + 2) + 2 y^{2} = 68$

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

Step 2.1.2.1.1.2

Apply the distributive property.

$2 y^{2} + 2 \cdot 2 + 2 y^{2} = 68$

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

Step 2.1.2.1.1.3

Multiply $2$ by $2$ .

$2 y^{2} + 4 + 2 y^{2} = 68$

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

$2 y^{2} + 4 + 2 y^{2} = 68$

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

Step 2.1.2.1.2

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

$4 y^{2} + 4 = 68$

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

$4 y^{2} + 4 = 68$

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

$4 y^{2} + 4 = 68$

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

$4 y^{2} + 4 = 68$

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

Step 2.2

Solve for $y$ in $4 y^{2} + 4 = 68$ .

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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.

$4 y^{2} = 68 - 4$

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

Step 2.2.1.2

Subtract $4$ from $68$ .

$4 y^{2} = 64$

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

$4 y^{2} = 64$

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

Step 2.2.2

Divide each term in $4 y^{2} = 64$ by $4$ and simplify.

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Step 2.2.2.1

Divide each term in $4 y^{2} = 64$ by $4$ .

$\frac{4 y^{2}}{4} = \frac{64}{4}$

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

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{4 y^{2}}{4} = \frac{64}{4}$

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

Step 2.2.2.2.1.2

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

$y^{2} = \frac{64}{4}$

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

$y^{2} = \frac{64}{4}$

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

$y^{2} = \frac{64}{4}$

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

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $64$ by $4$ .

$y^{2} = 16$

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

$y^{2} = 16$

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

$y^{2} = 16$

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

Step 2.2.3

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

$y = \pm \sqrt{16}$

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

Step 2.2.4

Simplify $\pm \sqrt{16}$ .

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Step 2.2.4.1

Rewrite $16$ as $4^{2}$ .

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

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

Step 2.2.4.2

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

$y = \pm 4$

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

$y = \pm 4$

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

Step 2.2.5

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

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Step 2.2.5.1

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

$y = 4$

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

Step 2.2.5.2

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

$y = - 4$

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

Step 2.2.5.3

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

$y = 4, - 4$

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

$y = 4, - 4$

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

$y = 4, - 4$

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

Step 2.3

Replace all occurrences of $y$ with $4$ in each equation.

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Step 2.3.1

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

$x = \sqrt{2 ({(4)}^{2} + 2)}$

$y = 4$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{2 ({(4)}^{2} + 2)}$ .

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Step 2.3.2.1.1

Raise $4$ to the power of $2$ .

$x = \sqrt{2 (16 + 2)}$

$y = 4$

Step 2.3.2.1.2

Add $16$ and $2$ .

$x = \sqrt{2 \cdot 18}$

$y = 4$

Step 2.3.2.1.3

Multiply $2$ by $18$ .

$x = \sqrt{36}$

$y = 4$

Step 2.3.2.1.4

Rewrite $36$ as $6^{2}$ .

$x = \sqrt{6^{2}}$

$y = 4$

Step 2.3.2.1.5

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

$x = 6$

$y = 4$

$x = 6$

$y = 4$

$x = 6$

$y = 4$

$x = 6$

$y = 4$

Step 2.4

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 2.4.1

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

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

$y = - 4$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{2 ({(- 4)}^{2} + 2)}$ .

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Step 2.4.2.1.1

Raise $- 4$ to the power of $2$ .

$x = \sqrt{2 (16 + 2)}$

$y = - 4$

Step 2.4.2.1.2

Add $16$ and $2$ .

$x = \sqrt{2 \cdot 18}$

$y = - 4$

Step 2.4.2.1.3

Multiply $2$ by $18$ .

$x = \sqrt{36}$

$y = - 4$

Step 2.4.2.1.4

Rewrite $36$ as $6^{2}$ .

$x = \sqrt{6^{2}}$

$y = - 4$

Step 2.4.2.1.5

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

$x = 6$

$y = - 4$

$x = 6$

$y = - 4$

$x = 6$

$y = - 4$

$x = 6$

$y = - 4$

$x = 6$

$y = - 4$

Step 3

Solve the system $x = - \sqrt{2 (y^{2} + 2)}, x^{2} + 2 y^{2} = 68$ .

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Step 3.1

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

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Step 3.1.1

Replace all occurrences of $x$ in $x^{2} + 2 y^{2} = 68$ with $- \sqrt{2 (y^{2} + 2)}$ .

${(- \sqrt{2 (y^{2} + 2)})}^{2} + 2 y^{2} = 68$

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

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

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

${(- 1)}^{2} {\sqrt{2 (y^{2} + 2)}}^{2} + 2 y^{2} = 68$

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

Step 3.1.2.1.1.2

Raise $- 1$ to the power of $2$ .

$1 {\sqrt{2 (y^{2} + 2)}}^{2} + 2 y^{2} = 68$

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

Step 3.1.2.1.1.3

Multiply ${\sqrt{2 (y^{2} + 2)}}^{2}$ by $1$ .

${\sqrt{2 (y^{2} + 2)}}^{2} + 2 y^{2} = 68$

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

Step 3.1.2.1.1.4

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

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Step 3.1.2.1.1.4.1

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

${({(2 (y^{2} + 2))}^{\frac{1}{2}})}^{2} + 2 y^{2} = 68$

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

Step 3.1.2.1.1.4.2

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

${(2 (y^{2} + 2))}^{\frac{1}{2} \cdot 2} + 2 y^{2} = 68$

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

Step 3.1.2.1.1.4.3

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

${(2 (y^{2} + 2))}^{\frac{2}{2}} + 2 y^{2} = 68$

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

Step 3.1.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.4.1

Cancel the common factor.

${(2 (y^{2} + 2))}^{\frac{2}{2}} + 2 y^{2} = 68$

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

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$(2 (y^{2} + 2)) + 2 y^{2} = 68$

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

$(2 (y^{2} + 2)) + 2 y^{2} = 68$

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

Step 3.1.2.1.1.4.5

Simplify.

$2 (y^{2} + 2) + 2 y^{2} = 68$

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

$2 (y^{2} + 2) + 2 y^{2} = 68$

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

Step 3.1.2.1.1.5

Apply the distributive property.

$2 y^{2} + 2 \cdot 2 + 2 y^{2} = 68$

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

Step 3.1.2.1.1.6

Multiply $2$ by $2$ .

$2 y^{2} + 4 + 2 y^{2} = 68$

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

$2 y^{2} + 4 + 2 y^{2} = 68$

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

Step 3.1.2.1.2

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

$4 y^{2} + 4 = 68$

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

$4 y^{2} + 4 = 68$

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

$4 y^{2} + 4 = 68$

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

$4 y^{2} + 4 = 68$

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

Step 3.2

Solve for $y$ in $4 y^{2} + 4 = 68$ .

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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.

$4 y^{2} = 68 - 4$

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

Step 3.2.1.2

Subtract $4$ from $68$ .

$4 y^{2} = 64$

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

$4 y^{2} = 64$

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

Step 3.2.2

Divide each term in $4 y^{2} = 64$ by $4$ and simplify.

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Step 3.2.2.1

Divide each term in $4 y^{2} = 64$ by $4$ .

$\frac{4 y^{2}}{4} = \frac{64}{4}$

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{4 y^{2}}{4} = \frac{64}{4}$

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

Step 3.2.2.2.1.2

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

$y^{2} = \frac{64}{4}$

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

$y^{2} = \frac{64}{4}$

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

$y^{2} = \frac{64}{4}$

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

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $64$ by $4$ .

$y^{2} = 16$

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

$y^{2} = 16$

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

$y^{2} = 16$

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

Step 3.2.3

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

$y = \pm \sqrt{16}$

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

Step 3.2.4

Simplify $\pm \sqrt{16}$ .

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Step 3.2.4.1

Rewrite $16$ as $4^{2}$ .

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

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

Step 3.2.4.2

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

$y = \pm 4$

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

$y = \pm 4$

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

Step 3.2.5

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

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Step 3.2.5.1

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

$y = 4$

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

Step 3.2.5.2

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

$y = - 4$

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

Step 3.2.5.3

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

$y = 4, - 4$

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

$y = 4, - 4$

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

$y = 4, - 4$

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

Step 3.3

Replace all occurrences of $y$ with $4$ in each equation.

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Step 3.3.1

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

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

$y = 4$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{2 ({(4)}^{2} + 2)}$ .

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Step 3.3.2.1.1

Raise $4$ to the power of $2$ .

$x = - \sqrt{2 (16 + 2)}$

$y = 4$

Step 3.3.2.1.2

Add $16$ and $2$ .

$x = - \sqrt{2 \cdot 18}$

$y = 4$

Step 3.3.2.1.3

Multiply $2$ by $18$ .

$x = - \sqrt{36}$

$y = 4$

Step 3.3.2.1.4

Rewrite $36$ as $6^{2}$ .

$x = - \sqrt{6^{2}}$

$y = 4$

Step 3.3.2.1.5

Multiply.

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Step 3.3.2.1.5.1

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

$x = - 1 \cdot 6$

$y = 4$

Step 3.3.2.1.5.2

Multiply $- 1$ by $6$ .

$x = - 6$

$y = 4$

$x = - 6$

$y = 4$

$x = - 6$

$y = 4$

$x = - 6$

$y = 4$

$x = - 6$

$y = 4$

Step 3.4

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 3.4.1

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

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

$y = - 4$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \sqrt{2 ({(- 4)}^{2} + 2)}$ .

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Step 3.4.2.1.1

Raise $- 4$ to the power of $2$ .

$x = - \sqrt{2 (16 + 2)}$

$y = - 4$

Step 3.4.2.1.2

Add $16$ and $2$ .

$x = - \sqrt{2 \cdot 18}$

$y = - 4$

Step 3.4.2.1.3

Multiply $2$ by $18$ .

$x = - \sqrt{36}$

$y = - 4$

Step 3.4.2.1.4

Rewrite $36$ as $6^{2}$ .

$x = - \sqrt{6^{2}}$

$y = - 4$

Step 3.4.2.1.5

Multiply.

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Step 3.4.2.1.5.1

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

$x = - 1 \cdot 6$

$y = - 4$

Step 3.4.2.1.5.2

Multiply $- 1$ by $6$ .

$x = - 6$

$y = - 4$

$x = - 6$

$y = - 4$

$x = - 6$

$y = - 4$

$x = - 6$

$y = - 4$

$x = - 6$

$y = - 4$

$x = - 6$

$y = - 4$

Step 4

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

$(6, 4)$

$(6, - 4)$

$(- 6, 4)$

$(- 6, - 4)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(6, 4), (6, - 4), (- 6, 4), (- 6, - 4)$

Equation Form:

$x = 6, y = 4$

$x = 6, y = - 4$

$x = - 6, y = 4$

$x = - 6, y = - 4$

Step 6