Precalculus Examples

$y - 7 = \frac{1}{4} \cdot {(x - 3)}^{2}$

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 $x$ .

$(0) - 7 = \frac{1}{4} \cdot {(x - 3)}^{2}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{4} \cdot {(x - 3)}^{2} = (0) - 7$ .

$\frac{1}{4} \cdot {(x - 3)}^{2} = (0) - 7$

Step 1.2.2

Multiply both sides of the equation by $4$ .

$4 (\frac{1}{4} \cdot {(x - 3)}^{2}) = 4 ((0) - 7)$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $4 (\frac{1}{4} \cdot {(x - 3)}^{2})$ .

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Step 1.2.3.1.1.1

Combine $\frac{1}{4}$ and ${(x - 3)}^{2}$ .

$4 \frac{{(x - 3)}^{2}}{4} = 4 ((0) - 7)$

Step 1.2.3.1.1.2

Cancel the common factor of $4$ .

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Step 1.2.3.1.1.2.1

Cancel the common factor.

$4 \frac{{(x - 3)}^{2}}{4} = 4 ((0) - 7)$

Step 1.2.3.1.1.2.2

Rewrite the expression.

${(x - 3)}^{2} = 4 ((0) - 7)$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Simplify $4 ((0) - 7)$ .

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Step 1.2.3.2.1.1

Subtract $7$ from $0$ .

${(x - 3)}^{2} = 4 \cdot - 7$

Step 1.2.3.2.1.2

Multiply $4$ by $- 7$ .

${(x - 3)}^{2} = - 28$

Step 1.2.4

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

$x - 3 = \pm \sqrt{- 28}$

Step 1.2.5

Simplify $\pm \sqrt{- 28}$ .

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Step 1.2.5.1

Rewrite $- 28$ as $- 1 (28)$ .

$x - 3 = \pm \sqrt{- 1 (28)}$

Step 1.2.5.2

Rewrite $\sqrt{- 1 (28)}$ as $\sqrt{- 1} \cdot \sqrt{28}$ .

$x - 3 = \pm \sqrt{- 1} \cdot \sqrt{28}$

Step 1.2.5.3

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

$x - 3 = \pm i \cdot \sqrt{28}$

Step 1.2.5.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 1.2.5.4.1

Factor $4$ out of $28$ .

$x - 3 = \pm i \cdot \sqrt{4 (7)}$

Step 1.2.5.4.2

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

$x - 3 = \pm i \cdot \sqrt{2^{2} \cdot 7}$

Step 1.2.5.5

Pull terms out from under the radical.

$x - 3 = \pm i \cdot (2 \sqrt{7})$

Step 1.2.5.6

Move $2$ to the left of $i$ .

$x - 3 = \pm 2 i \sqrt{7}$

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.

$x - 3 = 2 i \sqrt{7}$

Step 1.2.6.2

Add $3$ to both sides of the equation.

$x = 2 i \sqrt{7} + 3$

Step 1.2.6.3

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

$x - 3 = - 2 i \sqrt{7}$

Step 1.2.6.4

Add $3$ to both sides of the equation.

$x = - 2 i \sqrt{7} + 3$

Step 1.2.6.5

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

$x = 2 i \sqrt{7} + 3, - 2 i \sqrt{7} + 3$

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

$y - 7 = \frac{1}{4} \cdot {((0) - 3)}^{2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $\frac{1}{4} \cdot {((0) - 3)}^{2}$ .

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Step 2.2.1.1

Simplify the expression.

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Step 2.2.1.1.1

Subtract $3$ from $0$ .

$y - 7 = \frac{1}{4} \cdot {(- 3)}^{2}$

Step 2.2.1.1.2

Raise $- 3$ to the power of $2$ .

$y - 7 = \frac{1}{4} \cdot 9$

Step 2.2.1.2

Combine $\frac{1}{4}$ and $9$ .

$y - 7 = \frac{9}{4}$

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

Add $7$ to both sides of the equation.

$y = \frac{9}{4} + 7$

Step 2.2.2.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{9}{4} + 7 \cdot \frac{4}{4}$

Step 2.2.2.3

Combine $7$ and $\frac{4}{4}$ .

$y = \frac{9}{4} + \frac{7 \cdot 4}{4}$

Step 2.2.2.4

Combine the numerators over the common denominator.

$y = \frac{9 + 7 \cdot 4}{4}$

Step 2.2.2.5

Simplify the numerator.

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Step 2.2.2.5.1

Multiply $7$ by $4$ .

$y = \frac{9 + 28}{4}$

Step 2.2.2.5.2

Add $9$ and $28$ .

$y = \frac{37}{4}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{37}{4})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, \frac{37}{4})$

Step 4