Precalculus Examples

$3 {(x - 4)}^{2} + 3 y^{2} = 3$

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 $x$ .

$3 {(x - 4)}^{2} + 3 {(0)}^{2} = 3$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$3 {(x - 4)}^{2} = 3 - 3 {(0)}^{2}$

Step 1.2.2

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

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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 $0$ .

$3 {(x - 4)}^{2} = 3 - 3 \cdot 0$

Step 1.2.2.1.2

Multiply $- 3$ by $0$ .

$3 {(x - 4)}^{2} = 3 + 0$

Step 1.2.2.2

Add $3$ and $0$ .

$3 {(x - 4)}^{2} = 3$

Step 1.2.3

Divide each term in $3 {(x - 4)}^{2} = 3$ by $3$ and simplify.

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Step 1.2.3.1

Divide each term in $3 {(x - 4)}^{2} = 3$ by $3$ .

$\frac{3 {(x - 4)}^{2}}{3} = \frac{3}{3}$

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.

$\frac{3 {(x - 4)}^{2}}{3} = \frac{3}{3}$

Step 1.2.3.2.1.2

Divide ${(x - 4)}^{2}$ by $1$ .

${(x - 4)}^{2} = \frac{3}{3}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $3$ by $3$ .

${(x - 4)}^{2} = 1$

Step 1.2.4

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

$x - 4 = \pm \sqrt{1}$

Step 1.2.5

Any root of $1$ is $1$ .

$x - 4 = \pm 1$

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.

$x - 4 = 1$

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 $4$ to both sides of the equation.

$x = 1 + 4$

Step 1.2.6.2.2

Add $1$ and $4$ .

$x = 5$

Step 1.2.6.3

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

$x - 4 = - 1$

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 $4$ to both sides of the equation.

$x = - 1 + 4$

Step 1.2.6.4.2

Add $- 1$ and $4$ .

$x = 3$

Step 1.2.6.5

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

$x = 5, 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(5, 0), (3, 0)$

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 $y$ .

$3 {((0) - 4)}^{2} + 3 y^{2} = 3$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $3 {((0) - 4)}^{2} + 3 y^{2}$ .

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Step 2.2.1.1

Subtract $4$ from $0$ .

$3 {(- 4)}^{2} + 3 y^{2} = 3$

Step 2.2.1.2

Simplify each term.

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Step 2.2.1.2.1

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

$3 \cdot 16 + 3 y^{2} = 3$

Step 2.2.1.2.2

Multiply $3$ by $16$ .

$48 + 3 y^{2} = 3$

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 $48$ from both sides of the equation.

$3 y^{2} = 3 - 48$

Step 2.2.2.2

Subtract $48$ from $3$ .

$3 y^{2} = - 45$

Step 2.2.3

Divide each term in $3 y^{2} = - 45$ by $3$ and simplify.

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Step 2.2.3.1

Divide each term in $3 y^{2} = - 45$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{- 45}{3}$

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.

$\frac{3 y^{2}}{3} = \frac{- 45}{3}$

Step 2.2.3.2.1.2

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

$y^{2} = \frac{- 45}{3}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $- 45$ by $3$ .

$y^{2} = - 15$

Step 2.2.4

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

$y = \pm \sqrt{- 15}$

Step 2.2.5

Simplify $\pm \sqrt{- 15}$ .

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Step 2.2.5.1

Rewrite $- 15$ as $- 1 (15)$ .

$y = \pm \sqrt{- 1 (15)}$

Step 2.2.5.2

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$y = \pm \sqrt{- 1} \cdot \sqrt{15}$

Step 2.2.5.3

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

$y = \pm i \sqrt{15}$

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.

$y = i \sqrt{15}$

Step 2.2.6.2

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

$y = - i \sqrt{15}$

Step 2.2.6.3

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

$y = i \sqrt{15}, - i \sqrt{15}$

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): $(5, 0), (3, 0)$

y-intercept(s): $None$

Step 4