Precalculus Examples

$x^{2} + y^{2} = 9$ , $y = x^{2} - 3$

Step 1

Replace all occurrences of $y$ with $x^{2} - 3$ in each equation.

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Step 1.1

Replace all occurrences of $y$ in $x^{2} + y^{2} = 9$ with $x^{2} - 3$ .

$x^{2} + {(x^{2} - 3)}^{2} = 9$

$y = x^{2} - 3$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify $x^{2} + {(x^{2} - 3)}^{2}$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x^{2} - 3)}^{2}$ as $(x^{2} - 3) (x^{2} - 3)$ .

$x^{2} + (x^{2} - 3) (x^{2} - 3) = 9$

$y = x^{2} - 3$

Step 1.2.1.1.2

Expand $(x^{2} - 3) (x^{2} - 3)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$x^{2} + x^{2} (x^{2} - 3) - 3 (x^{2} - 3) = 9$

$y = x^{2} - 3$

Step 1.2.1.1.2.2

Apply the distributive property.

$x^{2} + x^{2} x^{2} + x^{2} \cdot - 3 - 3 (x^{2} - 3) = 9$

$y = x^{2} - 3$

Step 1.2.1.1.2.3

Apply the distributive property.

$x^{2} + x^{2} x^{2} + x^{2} \cdot - 3 - 3 x^{2} - 3 \cdot - 3 = 9$

$y = x^{2} - 3$

$x^{2} + x^{2} x^{2} + x^{2} \cdot - 3 - 3 x^{2} - 3 \cdot - 3 = 9$

$y = x^{2} - 3$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

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

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Step 1.2.1.1.3.1.1.1

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

$x^{2} + x^{2 + 2} + x^{2} \cdot - 3 - 3 x^{2} - 3 \cdot - 3 = 9$

$y = x^{2} - 3$

Step 1.2.1.1.3.1.1.2

Add $2$ and $2$ .

$x^{2} + x^{4} + x^{2} \cdot - 3 - 3 x^{2} - 3 \cdot - 3 = 9$

$y = x^{2} - 3$

$x^{2} + x^{4} + x^{2} \cdot - 3 - 3 x^{2} - 3 \cdot - 3 = 9$

$y = x^{2} - 3$

Step 1.2.1.1.3.1.2

Move $- 3$ to the left of $x^{2}$ .

$x^{2} + x^{4} - 3 \cdot x^{2} - 3 x^{2} - 3 \cdot - 3 = 9$

$y = x^{2} - 3$

Step 1.2.1.1.3.1.3

Multiply $- 3$ by $- 3$ .

$x^{2} + x^{4} - 3 x^{2} - 3 x^{2} + 9 = 9$

$y = x^{2} - 3$

$x^{2} + x^{4} - 3 x^{2} - 3 x^{2} + 9 = 9$

$y = x^{2} - 3$

Step 1.2.1.1.3.2

Subtract $3 x^{2}$ from $- 3 x^{2}$ .

$x^{2} + x^{4} - 6 x^{2} + 9 = 9$

$y = x^{2} - 3$

$x^{2} + x^{4} - 6 x^{2} + 9 = 9$

$y = x^{2} - 3$

$x^{2} + x^{4} - 6 x^{2} + 9 = 9$

$y = x^{2} - 3$

Step 1.2.1.2

Subtract $6 x^{2}$ from $x^{2}$ .

$x^{4} - 5 x^{2} + 9 = 9$

$y = x^{2} - 3$

$x^{4} - 5 x^{2} + 9 = 9$

$y = x^{2} - 3$

$x^{4} - 5 x^{2} + 9 = 9$

$y = x^{2} - 3$

$x^{4} - 5 x^{2} + 9 = 9$

$y = x^{2} - 3$

Step 2

Solve for $x$ in $x^{4} - 5 x^{2} + 9 = 9$ .

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Step 2.1

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 5 u + 9 = 9$

$u = x^{2}$

$y = x^{2} - 3$

Step 2.2

Subtract $9$ from both sides of the equation.

$u^{2} - 5 u + 9 - 9 = 0$

$y = x^{2} - 3$

Step 2.3

Combine the opposite terms in $u^{2} - 5 u + 9 - 9$ .

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Step 2.3.1

Subtract $9$ from $9$ .

$u^{2} - 5 u + 0 = 0$

$y = x^{2} - 3$

Step 2.3.2

Add $u^{2} - 5 u$ and $0$ .

$u^{2} - 5 u = 0$

$y = x^{2} - 3$

$u^{2} - 5 u = 0$

$y = x^{2} - 3$

Step 2.4

Factor $u$ out of $u^{2} - 5 u$ .

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Step 2.4.1

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

$u \cdot u - 5 u = 0$

$y = x^{2} - 3$

Step 2.4.2

Factor $u$ out of $- 5 u$ .

$u \cdot u + u \cdot - 5 = 0$

$y = x^{2} - 3$

Step 2.4.3

Factor $u$ out of $u \cdot u + u \cdot - 5$ .

$u (u - 5) = 0$

$y = x^{2} - 3$

$u (u - 5) = 0$

$y = x^{2} - 3$

Step 2.5

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

$u = 0$

$u - 5 = 0$

$y = x^{2} - 3$

Step 2.6

Set $u$ equal to $0$ .

$u = 0$

$y = x^{2} - 3$

Step 2.7

Set $u - 5$ equal to $0$ and solve for $u$ .

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Step 2.7.1

Set $u - 5$ equal to $0$ .

$u - 5 = 0$

$y = x^{2} - 3$

Step 2.7.2

Add $5$ to both sides of the equation.

$u = 5$

$y = x^{2} - 3$

$u = 5$

$y = x^{2} - 3$

Step 2.8

The final solution is all the values that make $u (u - 5) = 0$ true.

$u = 0, 5$

$y = x^{2} - 3$

Step 2.9

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 0$

$x^{2} = 5$

$y = x^{2} - 3$

Step 2.10

Solve the first equation for $x$ .

$x^{2} = 0$

$y = x^{2} - 3$

Step 2.11

Solve the equation for $x$ .

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Step 2.11.1

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

$x = \pm \sqrt{0}$

$y = x^{2} - 3$

Step 2.11.2

Simplify $\pm \sqrt{0}$ .

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Step 2.11.2.1

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

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

$y = x^{2} - 3$

Step 2.11.2.2

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

$x = \pm 0$

$y = x^{2} - 3$

Step 2.11.2.3

Plus or minus $0$ is $0$ .

$x = 0$

$y = x^{2} - 3$

$x = 0$

$y = x^{2} - 3$

$x = 0$

$y = x^{2} - 3$

Step 2.12

Solve the second equation for $x$ .

$x^{2} = 5$

$y = x^{2} - 3$

Step 2.13

Solve the equation for $x$ .

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Step 2.13.1

Remove parentheses.

$x^{2} = 5$

$y = x^{2} - 3$

Step 2.13.2

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

$x = \pm \sqrt{5}$

$y = x^{2} - 3$

Step 2.13.3

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

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Step 2.13.3.1

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

$x = \sqrt{5}$

$y = x^{2} - 3$

Step 2.13.3.2

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

$x = - \sqrt{5}$

$y = x^{2} - 3$

Step 2.13.3.3

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

$x = \sqrt{5}, - \sqrt{5}$

$y = x^{2} - 3$

$x = \sqrt{5}, - \sqrt{5}$

$y = x^{2} - 3$

$x = \sqrt{5}, - \sqrt{5}$

$y = x^{2} - 3$

Step 2.14

The solution to $x^{4} - 5 x^{2} + 9 = 9$ is $x = 0, \sqrt{5}, - \sqrt{5}$ .

$x = 0, \sqrt{5}, - \sqrt{5}$

$y = x^{2} - 3$

$x = 0, \sqrt{5}, - \sqrt{5}$

$y = x^{2} - 3$

Step 3

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

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Step 3.1

Replace all occurrences of $x$ in $y = x^{2} - 3$ with $0$ .

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

$x = 0$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify ${(0)}^{2} - 3$ .

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Step 3.2.1.1

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

$y = 0 - 3$

$x = 0$

Step 3.2.1.2

Subtract $3$ from $0$ .

$y = - 3$

$x = 0$

$y = - 3$

$x = 0$

$y = - 3$

$x = 0$

$y = - 3$

$x = 0$

Step 4

Replace all occurrences of $x$ with $\sqrt{5}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = x^{2} - 3$ with $\sqrt{5}$ .

$y = {(\sqrt{5})}^{2} - 3$

$x = \sqrt{5}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify ${(\sqrt{5})}^{2} - 3$ .

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Step 4.2.1.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$y = {(5^{\frac{1}{2}})}^{2} - 3$

$x = \sqrt{5}$

Step 4.2.1.1.2

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

$y = 5^{\frac{1}{2} \cdot 2} - 3$

$x = \sqrt{5}$

Step 4.2.1.1.3

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

$y = 5^{\frac{2}{2}} - 3$

$x = \sqrt{5}$

Step 4.2.1.1.4

Cancel the common factor of $2$ .

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Step 4.2.1.1.4.1

Cancel the common factor.

$y = 5^{\frac{2}{2}} - 3$

$x = \sqrt{5}$

Step 4.2.1.1.4.2

Rewrite the expression.

$y = 5 - 3$

$x = \sqrt{5}$

$y = 5 - 3$

$x = \sqrt{5}$

Step 4.2.1.1.5

Evaluate the exponent.

$y = 5 - 3$

$x = \sqrt{5}$

$y = 5 - 3$

$x = \sqrt{5}$

Step 4.2.1.2

Subtract $3$ from $5$ .

$y = 2$

$x = \sqrt{5}$

$y = 2$

$x = \sqrt{5}$

$y = 2$

$x = \sqrt{5}$

$y = 2$

$x = \sqrt{5}$

Step 5

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

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Step 5.1

Replace all occurrences of $x$ in $y = x^{2} - 3$ with $0$ .

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

$x = 0$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify ${(0)}^{2} - 3$ .

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Step 5.2.1.1

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

$y = 0 - 3$

$x = 0$

Step 5.2.1.2

Subtract $3$ from $0$ .

$y = - 3$

$x = 0$

$y = - 3$

$x = 0$

$y = - 3$

$x = 0$

$y = - 3$

$x = 0$

Step 6

Replace all occurrences of $x$ with $\sqrt{5}$ in each equation.

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Step 6.1

Replace all occurrences of $x$ in $y = x^{2} - 3$ with $\sqrt{5}$ .

$y = {(\sqrt{5})}^{2} - 3$

$x = \sqrt{5}$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify ${(\sqrt{5})}^{2} - 3$ .

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Step 6.2.1.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 6.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$y = {(5^{\frac{1}{2}})}^{2} - 3$

$x = \sqrt{5}$

Step 6.2.1.1.2

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

$y = 5^{\frac{1}{2} \cdot 2} - 3$

$x = \sqrt{5}$

Step 6.2.1.1.3

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

$y = 5^{\frac{2}{2}} - 3$

$x = \sqrt{5}$

Step 6.2.1.1.4

Cancel the common factor of $2$ .

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Step 6.2.1.1.4.1

Cancel the common factor.

$y = 5^{\frac{2}{2}} - 3$

$x = \sqrt{5}$

Step 6.2.1.1.4.2

Rewrite the expression.

$y = 5 - 3$

$x = \sqrt{5}$

$y = 5 - 3$

$x = \sqrt{5}$

Step 6.2.1.1.5

Evaluate the exponent.

$y = 5 - 3$

$x = \sqrt{5}$

$y = 5 - 3$

$x = \sqrt{5}$

Step 6.2.1.2

Subtract $3$ from $5$ .

$y = 2$

$x = \sqrt{5}$

$y = 2$

$x = \sqrt{5}$

$y = 2$

$x = \sqrt{5}$

$y = 2$

$x = \sqrt{5}$

Step 7

Replace all occurrences of $x$ with $- \sqrt{5}$ in each equation.

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Step 7.1

Replace all occurrences of $x$ in $y = x^{2} - 3$ with $- \sqrt{5}$ .

$y = {(- \sqrt{5})}^{2} - 3$

$x = - \sqrt{5}$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify ${(- \sqrt{5})}^{2} - 3$ .

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Step 7.2.1.1

Simplify each term.

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Step 7.2.1.1.1

Apply the product rule to $- \sqrt{5}$ .

$y = {(- 1)}^{2} {\sqrt{5}}^{2} - 3$

$x = - \sqrt{5}$

Step 7.2.1.1.2

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

$y = 1 {\sqrt{5}}^{2} - 3$

$x = - \sqrt{5}$

Step 7.2.1.1.3

Multiply ${\sqrt{5}}^{2}$ by $1$ .

$y = {\sqrt{5}}^{2} - 3$

$x = - \sqrt{5}$

Step 7.2.1.1.4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 7.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$y = {(5^{\frac{1}{2}})}^{2} - 3$

$x = - \sqrt{5}$

Step 7.2.1.1.4.2

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

$y = 5^{\frac{1}{2} \cdot 2} - 3$

$x = - \sqrt{5}$

Step 7.2.1.1.4.3

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

$y = 5^{\frac{2}{2}} - 3$

$x = - \sqrt{5}$

Step 7.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 7.2.1.1.4.4.1

Cancel the common factor.

$y = 5^{\frac{2}{2}} - 3$

$x = - \sqrt{5}$

Step 7.2.1.1.4.4.2

Rewrite the expression.

$y = 5 - 3$

$x = - \sqrt{5}$

$y = 5 - 3$

$x = - \sqrt{5}$

Step 7.2.1.1.4.5

Evaluate the exponent.

$y = 5 - 3$

$x = - \sqrt{5}$

$y = 5 - 3$

$x = - \sqrt{5}$

$y = 5 - 3$

$x = - \sqrt{5}$

Step 7.2.1.2

Subtract $3$ from $5$ .

$y = 2$

$x = - \sqrt{5}$

$y = 2$

$x = - \sqrt{5}$

$y = 2$

$x = - \sqrt{5}$

$y = 2$

$x = - \sqrt{5}$

Step 8

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

$(0, - 3)$

$(\sqrt{5}, 2)$

$(- \sqrt{5}, 2)$

Step 9

The result can be shown in multiple forms.

Point Form:

$(0, - 3), (\sqrt{5}, 2), (- \sqrt{5}, 2)$

Equation Form:

$x = 0, y = - 3$

$x = \sqrt{5}, y = 2$

$x = - \sqrt{5}, y = 2$

Step 10