Precalculus Examples

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

Step 1

Subtract $\frac{{(x - 2)}^{2}}{4}$ from both sides of the equation.

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

Step 2

Divide ${(y - 3)}^{2}$ by $1$ .

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

Step 3

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

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Step 3.1

Combine into one fraction.

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Step 3.1.1

Write $1$ as a fraction with a common denominator.

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

Step 3.1.2

Combine the numerators over the common denominator.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

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

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

Step 3.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x - 2$ .

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

Step 3.2.3

Simplify.

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Step 3.2.3.1

Subtract $2$ from $2$ .

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

Step 3.2.3.2

Add $x$ and $0$ .

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

Step 3.2.3.3

Apply the distributive property.

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

Step 3.2.3.4

Multiply $- 1$ by $- 2$ .

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

Step 3.2.3.5

Add $2$ and $2$ .

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

Step 3.3

Simplify with factoring out.

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Step 3.3.1

Factor $- 1$ out of $- x$ .

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

Step 3.3.2

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 3.3.3

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

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

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

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

Step 3.3.4.2

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Simplify $\pm \sqrt{- \frac{x (x - 4)}{4}}$ .

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Step 5.1

Rewrite $- \frac{x (x - 4)}{4}$ as ${(\frac{1^{2}}{2})}^{2} \cdot (- x (x - 4))$ .

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Step 5.1.1

Factor the perfect power $1^{2}$ out of $x (x - 4)$ .

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

Step 5.1.2

Factor the perfect power $2^{2}$ out of $4$ .

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

Step 5.1.3

Rearrange the fraction $\frac{1^{2} (x (x - 4))}{2^{2} \cdot 1}$ .

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

Step 5.1.4

Reorder $- 1$ and ${(\frac{1}{2})}^{2}$ .

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

Step 5.1.5

Rewrite $1$ as $1^{2}$ .

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

Step 5.1.6

Add parentheses.

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

Step 5.1.7

Add parentheses.

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

Step 5.2

Pull terms out from under the radical.

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

Step 5.3

One to any power is one.

$y - 3 = \pm \frac{1}{2} \sqrt{- x (x - 4)}$

Step 5.4

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

$y - 3 = \pm \frac{\sqrt{- x (x - 4)}}{2}$

Step 6

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

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Step 6.1

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

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

Step 6.2

Add $3$ to both sides of the equation.

$y = \frac{\sqrt{- x (x - 4)}}{2} + 3$

Step 6.3

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

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

Step 6.4

Add $3$ to both sides of the equation.

$y = - \frac{\sqrt{- x (x - 4)}}{2} + 3$

Step 6.5

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

$y = \frac{\sqrt{- x (x - 4)}}{2} + 3$

$y = - \frac{\sqrt{- x (x - 4)}}{2} + 3$

$y = \frac{\sqrt{- x (x - 4)}}{2} + 3$

$y = - \frac{\sqrt{- x (x - 4)}}{2} + 3$

Step 7

Set the radicand in $\sqrt{- x (x - 4)}$ greater than or equal to $0$ to find where the expression is defined.

$- x (x - 4) \geq 0$

Step 8

Solve for $x$ .

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Step 8.1

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

$x = 0$

$x - 4 = 0$

Step 8.2

Set $x$ equal to $0$ .

$x = 0$

Step 8.3

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 8.3.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 8.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 8.4

The final solution is all the values that make $- x (x - 4) \geq 0$ true.

$x = 0, 4$

Step 8.5

Use each root to create test intervals.

$x < 0$

$0 < x < 4$

$x > 4$

Step 8.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 8.6.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 8.6.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 8.6.1.2

Replace $x$ with $- 2$ in the original inequality.

$2 ((- 2) - 4) \geq 0$

Step 8.6.1.3

The left side $- 12$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.6.2

Test a value on the interval $0 < x < 4$ to see if it makes the inequality true.

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Step 8.6.2.1

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 8.6.2.2

Replace $x$ with $2$ in the original inequality.

$- 2 ((2) - 4) \geq 0$

Step 8.6.2.3

The left side $4$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.6.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 8.6.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 8.6.3.2

Replace $x$ with $6$ in the original inequality.

$- 6 ((6) - 4) \geq 0$

Step 8.6.3.3

The left side $- 12$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

Step 8.7

The solution consists of all of the true intervals.

$0 \leq x \leq 4$

Step 9

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[0, 4]$

Set-Builder Notation:

${x | 0 \leq x \leq 4}$

Step 10

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[2, 4]$

Set-Builder Notation:

${y | 2 \leq y \leq 4}$

Step 11

Determine the domain and range.

Domain: $[0, 4], {x | 0 \leq x \leq 4}$

Range: $[2, 4], {y | 2 \leq y \leq 4}$

Step 12