Precalculus Examples

$3 {(x - 5)}^{2} + 3 y^{2} = 12$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$3 {(x - 5)}^{2} + 3 {(0)}^{2} = 12$

Step 1.2

Solve the equation.

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Step 1.2.1

Subtract $3 {(0)}^{2}$ from both sides of the equation.

$3 {(x - 5)}^{2} = 12 - 3 {(0)}^{2}$

Step 1.2.2

Simplify $12 - 3 {(0)}^{2}$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

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

$3 {(x - 5)}^{2} = 12 - 3 \cdot 0$

Step 1.2.2.1.2

Multiply $- 3$ by $0$ .

$3 {(x - 5)}^{2} = 12 + 0$

Step 1.2.2.2

Add $12$ and $0$ .

$3 {(x - 5)}^{2} = 12$

Step 1.2.3

Divide each term in $3 {(x - 5)}^{2} = 12$ by $3$ and simplify.

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Step 1.2.3.1

Divide each term in $3 {(x - 5)}^{2} = 12$ by $3$ .

$\frac{3 {(x - 5)}^{2}}{3} = \frac{12}{3}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{3 {(x - 5)}^{2}}{3} = \frac{12}{3}$

Step 1.2.3.2.1.2

Divide ${(x - 5)}^{2}$ by $1$ .

${(x - 5)}^{2} = \frac{12}{3}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $12$ by $3$ .

${(x - 5)}^{2} = 4$

Step 1.2.4

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

$x - 5 = \pm \sqrt{4}$

Step 1.2.5

Simplify $\pm \sqrt{4}$ .

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Step 1.2.5.1

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

$x - 5 = \pm \sqrt{2^{2}}$

Step 1.2.5.2

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

$x - 5 = \pm 2$

Step 1.2.6

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

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Step 1.2.6.1

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

$x - 5 = 2$

Step 1.2.6.2

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

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Step 1.2.6.2.1

Add $5$ to both sides of the equation.

$x = 2 + 5$

Step 1.2.6.2.2

Add $2$ and $5$ .

$x = 7$

Step 1.2.6.3

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

$x - 5 = - 2$

Step 1.2.6.4

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

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Step 1.2.6.4.1

Add $5$ to both sides of the equation.

$x = - 2 + 5$

Step 1.2.6.4.2

Add $- 2$ and $5$ .

$x = 3$

Step 1.2.6.5

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

$x = 7, 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(7, 0), (3, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$3 {((0) - 5)}^{2} + 3 y^{2} = 12$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $3 {((0) - 5)}^{2} + 3 y^{2}$ .

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Step 2.2.1.1

Subtract $5$ from $0$ .

$3 {(- 5)}^{2} + 3 y^{2} = 12$

Step 2.2.1.2

Simplify each term.

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Step 2.2.1.2.1

Raise $- 5$ to the power of $2$ .

$3 \cdot 25 + 3 y^{2} = 12$

Step 2.2.1.2.2

Multiply $3$ by $25$ .

$75 + 3 y^{2} = 12$

Step 2.2.2

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

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Step 2.2.2.1

Subtract $75$ from both sides of the equation.

$3 y^{2} = 12 - 75$

Step 2.2.2.2

Subtract $75$ from $12$ .

$3 y^{2} = - 63$

Step 2.2.3

Divide each term in $3 y^{2} = - 63$ by $3$ and simplify.

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Step 2.2.3.1

Divide each term in $3 y^{2} = - 63$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{- 63}{3}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $3$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{- 63}{3}$

Step 2.2.3.2.1.2

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

$y^{2} = \frac{- 63}{3}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $- 63$ by $3$ .

$y^{2} = - 21$

Step 2.2.4

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

$y = \pm \sqrt{- 21}$

Step 2.2.5

Simplify $\pm \sqrt{- 21}$ .

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Step 2.2.5.1

Rewrite $- 21$ as $- 1 (21)$ .

$y = \pm \sqrt{- 1 (21)}$

Step 2.2.5.2

Rewrite $\sqrt{- 1 (21)}$ as $\sqrt{- 1} \cdot \sqrt{21}$ .

$y = \pm \sqrt{- 1} \cdot \sqrt{21}$

Step 2.2.5.3

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \pm i \sqrt{21}$

Step 2.2.6

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

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Step 2.2.6.1

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

$y = i \sqrt{21}$

Step 2.2.6.2

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

$y = - i \sqrt{21}$

Step 2.2.6.3

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

$y = i \sqrt{21}, - i \sqrt{21}$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(7, 0), (3, 0)$

y-intercept(s): $None$

Step 4