Precalculus Examples

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

Step 1

Replace all occurrences of $x$ with $y^{2}$ in each equation.

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Step 1.1

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

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

$x = y^{2}$

Step 1.2

Simplify ${(y^{2})}^{2} + y^{2} = 4 (y^{2})$ .

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Step 1.2.1

Simplify the left side.

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Step 1.2.1.1

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 1.2.1.1.1

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

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

$x = y^{2}$

Step 1.2.1.1.2

Multiply $2$ by $2$ .

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

$x = y^{2}$

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

$x = y^{2}$

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

$x = y^{2}$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

Multiply $4$ by $y^{2}$ .

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

$x = y^{2}$

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

$x = y^{2}$

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

$x = y^{2}$

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

$x = y^{2}$

Step 2

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

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Step 2.1

Move all terms containing $y$ to the left side of the equation.

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Step 2.1.1

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

$y^{4} + y^{2} - 4 y^{2} = 0$

$x = y^{2}$

Step 2.1.2

Subtract $4 y^{2}$ from $y^{2}$ .

$y^{4} - 3 y^{2} = 0$

$x = y^{2}$

$y^{4} - 3 y^{2} = 0$

$x = y^{2}$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

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

${(y^{2})}^{2} - 3 y^{2} = 0$

$x = y^{2}$

Step 2.2.2

Let $u = y^{2}$ . Substitute $u$ for all occurrences of $y^{2}$ .

$u^{2} - 3 u = 0$

$x = y^{2}$

Step 2.2.3

Factor $u$ out of $u^{2} - 3 u$ .

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Step 2.2.3.1

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

$u \cdot u - 3 u = 0$

$x = y^{2}$

Step 2.2.3.2

Factor $u$ out of $- 3 u$ .

$u \cdot u + u \cdot - 3 = 0$

$x = y^{2}$

Step 2.2.3.3

Factor $u$ out of $u \cdot u + u \cdot - 3$ .

$u (u - 3) = 0$

$x = y^{2}$

$u (u - 3) = 0$

$x = y^{2}$

Step 2.2.4

Replace all occurrences of $u$ with $y^{2}$ .

$y^{2} (y^{2} - 3) = 0$

$x = y^{2}$

$y^{2} (y^{2} - 3) = 0$

$x = y^{2}$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y^{2} = 0$

$y^{2} - 3 = 0$

$x = y^{2}$

Step 2.4

Set $y^{2}$ equal to $0$ and solve for $y$ .

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Step 2.4.1

Set $y^{2}$ equal to $0$ .

$y^{2} = 0$

$x = y^{2}$

Step 2.4.2

Solve $y^{2} = 0$ for $y$ .

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Step 2.4.2.1

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

$y = \pm \sqrt{0}$

$x = y^{2}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.4.2.2.1

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

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

$x = y^{2}$

Step 2.4.2.2.2

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

$y = \pm 0$

$x = y^{2}$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$y = 0$

$x = y^{2}$

$y = 0$

$x = y^{2}$

$y = 0$

$x = y^{2}$

$y = 0$

$x = y^{2}$

Step 2.5

Set $y^{2} - 3$ equal to $0$ and solve for $y$ .

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Step 2.5.1

Set $y^{2} - 3$ equal to $0$ .

$y^{2} - 3 = 0$

$x = y^{2}$

Step 2.5.2

Solve $y^{2} - 3 = 0$ for $y$ .

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Step 2.5.2.1

Add $3$ to both sides of the equation.

$y^{2} = 3$

$x = y^{2}$

Step 2.5.2.2

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

$y = \pm \sqrt{3}$

$x = y^{2}$

Step 2.5.2.3

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

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Step 2.5.2.3.1

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

$y = \sqrt{3}$

$x = y^{2}$

Step 2.5.2.3.2

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

$y = - \sqrt{3}$

$x = y^{2}$

Step 2.5.2.3.3

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

$y = \sqrt{3}, - \sqrt{3}$

$x = y^{2}$

$y = \sqrt{3}, - \sqrt{3}$

$x = y^{2}$

$y = \sqrt{3}, - \sqrt{3}$

$x = y^{2}$

$y = \sqrt{3}, - \sqrt{3}$

$x = y^{2}$

Step 2.6

The final solution is all the values that make $y^{2} (y^{2} - 3) = 0$ true.

$y = 0, \sqrt{3}, - \sqrt{3}$

$x = y^{2}$

$y = 0, \sqrt{3}, - \sqrt{3}$

$x = y^{2}$

Step 3

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

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Step 3.1

Replace all occurrences of $y$ in $x = y^{2}$ with $0$ .

$x = {(0)}^{2}$

$y = 0$

Step 3.2

Simplify the right side.

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Step 3.2.1

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

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

Step 4

Replace all occurrences of $y$ with $\sqrt{3}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = y^{2}$ with $\sqrt{3}$ .

$x = {(\sqrt{3})}^{2}$

$y = \sqrt{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = {(3^{\frac{1}{2}})}^{2}$

$y = \sqrt{3}$

Step 4.2.1.2

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

$x = 3^{\frac{1}{2} \cdot 2}$

$y = \sqrt{3}$

Step 4.2.1.3

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

$x = 3^{\frac{2}{2}}$

$y = \sqrt{3}$

Step 4.2.1.4

Cancel the common factor of $2$ .

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Step 4.2.1.4.1

Cancel the common factor.

$x = 3^{\frac{2}{2}}$

$y = \sqrt{3}$

Step 4.2.1.4.2

Rewrite the expression.

$x = 3$

$y = \sqrt{3}$

$x = 3$

$y = \sqrt{3}$

Step 4.2.1.5

Evaluate the exponent.

$x = 3$

$y = \sqrt{3}$

$x = 3$

$y = \sqrt{3}$

$x = 3$

$y = \sqrt{3}$

$x = 3$

$y = \sqrt{3}$

Step 5

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

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Step 5.1

Replace all occurrences of $y$ in $x = y^{2}$ with $0$ .

$x = {(0)}^{2}$

$y = 0$

Step 5.2

Simplify the right side.

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Step 5.2.1

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

$x = 0$

$y = 0$

$x = 0$

$y = 0$

$x = 0$

$y = 0$

Step 6

Replace all occurrences of $y$ with $\sqrt{3}$ in each equation.

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Step 6.1

Replace all occurrences of $y$ in $x = y^{2}$ with $\sqrt{3}$ .

$x = {(\sqrt{3})}^{2}$

$y = \sqrt{3}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = {(3^{\frac{1}{2}})}^{2}$

$y = \sqrt{3}$

Step 6.2.1.2

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

$x = 3^{\frac{1}{2} \cdot 2}$

$y = \sqrt{3}$

Step 6.2.1.3

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

$x = 3^{\frac{2}{2}}$

$y = \sqrt{3}$

Step 6.2.1.4

Cancel the common factor of $2$ .

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Step 6.2.1.4.1

Cancel the common factor.

$x = 3^{\frac{2}{2}}$

$y = \sqrt{3}$

Step 6.2.1.4.2

Rewrite the expression.

$x = 3$

$y = \sqrt{3}$

$x = 3$

$y = \sqrt{3}$

Step 6.2.1.5

Evaluate the exponent.

$x = 3$

$y = \sqrt{3}$

$x = 3$

$y = \sqrt{3}$

$x = 3$

$y = \sqrt{3}$

$x = 3$

$y = \sqrt{3}$

Step 7

Replace all occurrences of $y$ with $- \sqrt{3}$ in each equation.

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Step 7.1

Replace all occurrences of $y$ in $x = y^{2}$ with $- \sqrt{3}$ .

$x = {(- \sqrt{3})}^{2}$

$y = - \sqrt{3}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify ${(- \sqrt{3})}^{2}$ .

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Step 7.2.1.1

Simplify the expression.

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Step 7.2.1.1.1

Apply the product rule to $- \sqrt{3}$ .

$x = {(- 1)}^{2} {\sqrt{3}}^{2}$

$y = - \sqrt{3}$

Step 7.2.1.1.2

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

$x = 1 {\sqrt{3}}^{2}$

$y = - \sqrt{3}$

Step 7.2.1.1.3

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$x = {\sqrt{3}}^{2}$

$y = - \sqrt{3}$

$x = {\sqrt{3}}^{2}$

$y = - \sqrt{3}$

Step 7.2.1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = {(3^{\frac{1}{2}})}^{2}$

$y = - \sqrt{3}$

Step 7.2.1.2.2

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

$x = 3^{\frac{1}{2} \cdot 2}$

$y = - \sqrt{3}$

Step 7.2.1.2.3

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

$x = 3^{\frac{2}{2}}$

$y = - \sqrt{3}$

Step 7.2.1.2.4

Cancel the common factor of $2$ .

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Step 7.2.1.2.4.1

Cancel the common factor.

$x = 3^{\frac{2}{2}}$

$y = - \sqrt{3}$

Step 7.2.1.2.4.2

Rewrite the expression.

$x = 3$

$y = - \sqrt{3}$

$x = 3$

$y = - \sqrt{3}$

Step 7.2.1.2.5

Evaluate the exponent.

$x = 3$

$y = - \sqrt{3}$

$x = 3$

$y = - \sqrt{3}$

$x = 3$

$y = - \sqrt{3}$

$x = 3$

$y = - \sqrt{3}$

$x = 3$

$y = - \sqrt{3}$

Step 8

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

$(0, 0)$

$(3, \sqrt{3})$

$(3, - \sqrt{3})$

Step 9

The result can be shown in multiple forms.

Point Form:

$(0, 0), (3, \sqrt{3}), (3, - \sqrt{3})$

Equation Form:

$x = 0, y = 0$

$x = 3, y = \sqrt{3}$

$x = 3, y = - \sqrt{3}$

Step 10