Precalculus Examples

$4 x^{2} + 9 y^{2} - 18 y - 27 = 0$

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 $x$ .

$4 x^{2} + 9 {(0)}^{2} - 18 \cdot 0 - 27 = 0$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $4 x^{2} + 9 {(0)}^{2} - 18 \cdot 0 - 27$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

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

$4 x^{2} + 9 \cdot 0 - 18 \cdot 0 - 27 = 0$

Step 1.2.1.1.2

Multiply $9$ by $0$ .

$4 x^{2} + 0 - 18 \cdot 0 - 27 = 0$

Step 1.2.1.1.3

Multiply $- 18$ by $0$ .

$4 x^{2} + 0 + 0 - 27 = 0$

Step 1.2.1.2

Combine the opposite terms in $4 x^{2} + 0 + 0 - 27$ .

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Step 1.2.1.2.1

Add $4 x^{2}$ and $0$ .

$4 x^{2} + 0 - 27 = 0$

Step 1.2.1.2.2

Add $4 x^{2}$ and $0$ .

$4 x^{2} - 27 = 0$

Step 1.2.2

Add $27$ to both sides of the equation.

$4 x^{2} = 27$

Step 1.2.3

Divide each term in $4 x^{2} = 27$ by $4$ and simplify.

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Step 1.2.3.1

Divide each term in $4 x^{2} = 27$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{27}{4}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $4$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{27}{4}$

Step 1.2.3.2.1.2

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

$x^{2} = \frac{27}{4}$

Step 1.2.4

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

$x = \pm \sqrt{\frac{27}{4}}$

Step 1.2.5

Simplify $\pm \sqrt{\frac{27}{4}}$ .

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Step 1.2.5.1

Rewrite $\sqrt{\frac{27}{4}}$ as $\frac{\sqrt{27}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{27}}{\sqrt{4}}$

Step 1.2.5.2

Simplify the numerator.

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Step 1.2.5.2.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.2.5.2.1.1

Factor $9$ out of $27$ .

$x = \pm \frac{\sqrt{9 (3)}}{\sqrt{4}}$

Step 1.2.5.2.1.2

Rewrite $9$ as $3^{2}$ .

$x = \pm \frac{\sqrt{3^{2} \cdot 3}}{\sqrt{4}}$

Step 1.2.5.2.2

Pull terms out from under the radical.

$x = \pm \frac{3 \sqrt{3}}{\sqrt{4}}$

Step 1.2.5.3

Simplify the denominator.

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Step 1.2.5.3.1

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

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

Step 1.2.5.3.2

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

$x = \pm \frac{3 \sqrt{3}}{2}$

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.

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

Step 1.2.6.2

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

$x = - \frac{3 \sqrt{3}}{2}$

Step 1.2.6.3

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

$x = \frac{3 \sqrt{3}}{2}, - \frac{3 \sqrt{3}}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{3 \sqrt{3}}{2}, 0), (- \frac{3 \sqrt{3}}{2}, 0)$

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 $y$ .

$4 {(0)}^{2} + 9 y^{2} - 18 y - 27 = 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $4 {(0)}^{2} + 9 y^{2} - 18 y - 27$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

$4 \cdot 0 + 9 y^{2} - 18 y - 27 = 0$

Step 2.2.1.1.2

Multiply $4$ by $0$ .

$0 + 9 y^{2} - 18 y - 27 = 0$

Step 2.2.1.2

Add $0$ and $9 y^{2}$ .

$9 y^{2} - 18 y - 27 = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $9$ out of $9 y^{2} - 18 y - 27$ .

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Step 2.2.2.1.1

Factor $9$ out of $9 y^{2}$ .

$9 (y^{2}) - 18 y - 27 = 0$

Step 2.2.2.1.2

Factor $9$ out of $- 18 y$ .

$9 (y^{2}) + 9 (- 2 y) - 27 = 0$

Step 2.2.2.1.3

Factor $9$ out of $- 27$ .

$9 y^{2} + 9 (- 2 y) + 9 \cdot - 3 = 0$

Step 2.2.2.1.4

Factor $9$ out of $9 y^{2} + 9 (- 2 y)$ .

$9 (y^{2} - 2 y) + 9 \cdot - 3 = 0$

Step 2.2.2.1.5

Factor $9$ out of $9 (y^{2} - 2 y) + 9 \cdot - 3$ .

$9 (y^{2} - 2 y - 3) = 0$

Step 2.2.2.2

Factor.

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Step 2.2.2.2.1

Factor $y^{2} - 2 y - 3$ using the AC method.

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Step 2.2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 2.2.2.2.1.2

Write the factored form using these integers.

$9 ((y - 3) (y + 1)) = 0$

Step 2.2.2.2.2

Remove unnecessary parentheses.

$9 (y - 3) (y + 1) = 0$

Step 2.2.3

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

$y - 3 = 0$

$y + 1 = 0$

Step 2.2.4

Set $y - 3$ equal to $0$ and solve for $y$ .

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Step 2.2.4.1

Set $y - 3$ equal to $0$ .

$y - 3 = 0$

Step 2.2.4.2

Add $3$ to both sides of the equation.

$y = 3$

Step 2.2.5

Set $y + 1$ equal to $0$ and solve for $y$ .

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Step 2.2.5.1

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Step 2.2.5.2

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 2.2.6

The final solution is all the values that make $9 (y - 3) (y + 1) = 0$ true.

$y = 3, - 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 3), (0, - 1)$

Step 3

List the intersections.

x-intercept(s): $(\frac{3 \sqrt{3}}{2}, 0), (- \frac{3 \sqrt{3}}{2}, 0)$

y-intercept(s): $(0, 3), (0, - 1)$

Step 4