Precalculus Examples

$4 x^{2} + 4 y^{2} - 16 x + 24 y + 16 = 0$

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 $x$ .

$4 x^{2} + 4 {(0)}^{2} - 16 x + 24 (0) + 16 = 0$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $4 x^{2} + 4 {(0)}^{2} - 16 x + 24 (0) + 16$ .

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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 $0$ .

$4 x^{2} + 4 \cdot 0 - 16 x + 24 (0) + 16 = 0$

Step 1.2.1.1.2

Multiply $4$ by $0$ .

$4 x^{2} + 0 - 16 x + 24 (0) + 16 = 0$

Step 1.2.1.1.3

Multiply $24$ by $0$ .

$4 x^{2} + 0 - 16 x + 0 + 16 = 0$

Step 1.2.1.2

Combine the opposite terms in $4 x^{2} + 0 - 16 x + 0 + 16$ .

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Step 1.2.1.2.1

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

$4 x^{2} - 16 x + 0 + 16 = 0$

Step 1.2.1.2.2

Add $4 x^{2} - 16 x$ and $0$ .

$4 x^{2} - 16 x + 16 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $4$ out of $4 x^{2} - 16 x + 16$ .

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Step 1.2.2.1.1

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) - 16 x + 16 = 0$

Step 1.2.2.1.2

Factor $4$ out of $- 16 x$ .

$4 (x^{2}) + 4 (- 4 x) + 16 = 0$

Step 1.2.2.1.3

Factor $4$ out of $16$ .

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

Step 1.2.2.1.4

Factor $4$ out of $4 x^{2} + 4 (- 4 x)$ .

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

Step 1.2.2.1.5

Factor $4$ out of $4 (x^{2} - 4 x) + 4 \cdot 4$ .

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

Step 1.2.2.2

Factor using the perfect square rule.

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Step 1.2.2.2.1

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

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

Step 1.2.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 1.2.2.2.3

Rewrite the polynomial.

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

Step 1.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 1.2.3

Divide each term in $4 {(x - 2)}^{2} = 0$ by $4$ and simplify.

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Step 1.2.3.1

Divide each term in $4 {(x - 2)}^{2} = 0$ by $4$ .

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

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.

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

Step 1.2.3.2.1.2

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

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

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $4$ .

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

Step 1.2.4

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.5

Add $2$ to both sides of the equation.

$x = 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0)$

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 $y$ .

$4 {(0)}^{2} + 4 y^{2} - 16 \cdot 0 + 24 y + 16 = 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $4 {(0)}^{2} + 4 y^{2} - 16 \cdot 0 + 24 y + 16$ .

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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 $0$ .

$4 \cdot 0 + 4 y^{2} - 16 \cdot 0 + 24 y + 16 = 0$

Step 2.2.1.1.2

Multiply $4$ by $0$ .

$0 + 4 y^{2} - 16 \cdot 0 + 24 y + 16 = 0$

Step 2.2.1.1.3

Multiply $- 16$ by $0$ .

$0 + 4 y^{2} + 0 + 24 y + 16 = 0$

Step 2.2.1.2

Combine the opposite terms in $0 + 4 y^{2} + 0 + 24 y + 16$ .

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Step 2.2.1.2.1

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

$4 y^{2} + 0 + 24 y + 16 = 0$

Step 2.2.1.2.2

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

$4 y^{2} + 24 y + 16 = 0$

Step 2.2.2

Factor $4$ out of $4 y^{2} + 24 y + 16$ .

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Step 2.2.2.1

Factor $4$ out of $4 y^{2}$ .

$4 (y^{2}) + 24 y + 16 = 0$

Step 2.2.2.2

Factor $4$ out of $24 y$ .

$4 (y^{2}) + 4 (6 y) + 16 = 0$

Step 2.2.2.3

Factor $4$ out of $16$ .

$4 y^{2} + 4 (6 y) + 4 \cdot 4 = 0$

Step 2.2.2.4

Factor $4$ out of $4 y^{2} + 4 (6 y)$ .

$4 (y^{2} + 6 y) + 4 \cdot 4 = 0$

Step 2.2.2.5

Factor $4$ out of $4 (y^{2} + 6 y) + 4 \cdot 4$ .

$4 (y^{2} + 6 y + 4) = 0$

Step 2.2.3

Divide each term in $4 (y^{2} + 6 y + 4) = 0$ by $4$ and simplify.

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Step 2.2.3.1

Divide each term in $4 (y^{2} + 6 y + 4) = 0$ by $4$ .

$\frac{4 (y^{2} + 6 y + 4)}{4} = \frac{0}{4}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $4$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{4 (y^{2} + 6 y + 4)}{4} = \frac{0}{4}$

Step 2.2.3.2.1.2

Divide $y^{2} + 6 y + 4$ by $1$ .

$y^{2} + 6 y + 4 = \frac{0}{4}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $0$ by $4$ .

$y^{2} + 6 y + 4 = 0$

Step 2.2.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.2.5

Substitute the values $a = 1$ , $b = 6$ , and $c = 4$ into the quadratic formula and solve for $y$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 2.2.6

Simplify.

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Step 2.2.6.1

Simplify the numerator.

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Step 2.2.6.1.1

Raise $6$ to the power of $2$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 2.2.6.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 2.2.6.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 4}}{2 \cdot 1}$

Step 2.2.6.1.2.2

Multiply $- 4$ by $4$ .

$y = \frac{- 6 \pm \sqrt{36 - 16}}{2 \cdot 1}$

Step 2.2.6.1.3

Subtract $16$ from $36$ .

$y = \frac{- 6 \pm \sqrt{20}}{2 \cdot 1}$

Step 2.2.6.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.2.6.1.4.1

Factor $4$ out of $20$ .

$y = \frac{- 6 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 2.2.6.1.4.2

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

$y = \frac{- 6 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 2.2.6.1.5

Pull terms out from under the radical.

$y = \frac{- 6 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 2.2.6.2

Multiply $2$ by $1$ .

$y = \frac{- 6 \pm 2 \sqrt{5}}{2}$

Step 2.2.6.3

Simplify $\frac{- 6 \pm 2 \sqrt{5}}{2}$ .

$y = - 3 \pm \sqrt{5}$

Step 2.2.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.2.7.1

Simplify the numerator.

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Step 2.2.7.1.1

Raise $6$ to the power of $2$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 2.2.7.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 2.2.7.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 4}}{2 \cdot 1}$

Step 2.2.7.1.2.2

Multiply $- 4$ by $4$ .

$y = \frac{- 6 \pm \sqrt{36 - 16}}{2 \cdot 1}$

Step 2.2.7.1.3

Subtract $16$ from $36$ .

$y = \frac{- 6 \pm \sqrt{20}}{2 \cdot 1}$

Step 2.2.7.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.2.7.1.4.1

Factor $4$ out of $20$ .

$y = \frac{- 6 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 2.2.7.1.4.2

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

$y = \frac{- 6 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 2.2.7.1.5

Pull terms out from under the radical.

$y = \frac{- 6 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 2.2.7.2

Multiply $2$ by $1$ .

$y = \frac{- 6 \pm 2 \sqrt{5}}{2}$

Step 2.2.7.3

Simplify $\frac{- 6 \pm 2 \sqrt{5}}{2}$ .

$y = - 3 \pm \sqrt{5}$

Step 2.2.7.4

Change the $\pm$ to $+$ .

$y = - 3 + \sqrt{5}$

Step 2.2.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.2.8.1

Simplify the numerator.

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Step 2.2.8.1.1

Raise $6$ to the power of $2$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 2.2.8.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 2.2.8.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 4}}{2 \cdot 1}$

Step 2.2.8.1.2.2

Multiply $- 4$ by $4$ .

$y = \frac{- 6 \pm \sqrt{36 - 16}}{2 \cdot 1}$

Step 2.2.8.1.3

Subtract $16$ from $36$ .

$y = \frac{- 6 \pm \sqrt{20}}{2 \cdot 1}$

Step 2.2.8.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.2.8.1.4.1

Factor $4$ out of $20$ .

$y = \frac{- 6 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 2.2.8.1.4.2

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

$y = \frac{- 6 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 2.2.8.1.5

Pull terms out from under the radical.

$y = \frac{- 6 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 2.2.8.2

Multiply $2$ by $1$ .

$y = \frac{- 6 \pm 2 \sqrt{5}}{2}$

Step 2.2.8.3

Simplify $\frac{- 6 \pm 2 \sqrt{5}}{2}$ .

$y = - 3 \pm \sqrt{5}$

Step 2.2.8.4

Change the $\pm$ to $-$ .

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

Step 2.2.9

The final answer is the combination of both solutions.

$y = - 3 + \sqrt{5}, - 3 - \sqrt{5}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 3 + \sqrt{5}), (0, - 3 - \sqrt{5})$

Step 3

List the intersections.

x-intercept(s): $(2, 0)$

y-intercept(s): $(0, - 3 + \sqrt{5}), (0, - 3 - \sqrt{5})$

Step 4