Precalculus Examples

$\frac{{(y - 7)}^{2}}{81} - \frac{{(x - 3)}^{2}}{144} = 1$

Step 1

Add $\frac{{(x - 3)}^{2}}{144}$ to both sides of the equation.

$\frac{{(y - 7)}^{2}}{81} = 1 + \frac{{(x - 3)}^{2}}{144}$

Step 2

Simplify $1 + \frac{{(x - 3)}^{2}}{144}$ .

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Step 2.1

Combine into one fraction.

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Step 2.1.1

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

$\frac{{(y - 7)}^{2}}{81} = \frac{144}{144} + \frac{{(x - 3)}^{2}}{144}$

Step 2.1.2

Combine the numerators over the common denominator.

$\frac{{(y - 7)}^{2}}{81} = \frac{144 + {(x - 3)}^{2}}{144}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$\frac{{(y - 7)}^{2}}{81} = \frac{144 + (x - 3) (x - 3)}{144}$

Step 2.2.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

$\frac{{(y - 7)}^{2}}{81} = \frac{144 + x (x - 3) - 3 (x - 3)}{144}$

Step 2.2.2.2

Apply the distributive property.

$\frac{{(y - 7)}^{2}}{81} = \frac{144 + x \cdot x + x \cdot - 3 - 3 (x - 3)}{144}$

Step 2.2.2.3

Apply the distributive property.

$\frac{{(y - 7)}^{2}}{81} = \frac{144 + x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3}{144}$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $x$ by $x$ .

$\frac{{(y - 7)}^{2}}{81} = \frac{144 + x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3}{144}$

Step 2.2.3.1.2

Move $- 3$ to the left of $x$ .

$\frac{{(y - 7)}^{2}}{81} = \frac{144 + x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3}{144}$

Step 2.2.3.1.3

Multiply $- 3$ by $- 3$ .

$\frac{{(y - 7)}^{2}}{81} = \frac{144 + x^{2} - 3 x - 3 x + 9}{144}$

Step 2.2.3.2

Subtract $3 x$ from $- 3 x$ .

$\frac{{(y - 7)}^{2}}{81} = \frac{144 + x^{2} - 6 x + 9}{144}$

Step 2.2.4

Add $144$ and $9$ .

$\frac{{(y - 7)}^{2}}{81} = \frac{x^{2} - 6 x + 153}{144}$

Step 3

Multiply both sides of the equation by $81$ .

$81 \frac{{(y - 7)}^{2}}{81} = 81 \frac{x^{2} - 6 x + 153}{144}$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $81$ .

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Step 4.1.1.1

Cancel the common factor.

$81 \frac{{(y - 7)}^{2}}{81} = 81 \frac{x^{2} - 6 x + 153}{144}$

Step 4.1.1.2

Rewrite the expression.

${(y - 7)}^{2} = 81 \frac{x^{2} - 6 x + 153}{144}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $81 \frac{x^{2} - 6 x + 153}{144}$ .

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Step 4.2.1.1

Cancel the common factor of $9$ .

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Step 4.2.1.1.1

Factor $9$ out of $81$ .

${(y - 7)}^{2} = 9 (9) \frac{x^{2} - 6 x + 153}{144}$

Step 4.2.1.1.2

Factor $9$ out of $144$ .

${(y - 7)}^{2} = 9 \cdot 9 \frac{x^{2} - 6 x + 153}{9 \cdot 16}$

Step 4.2.1.1.3

Cancel the common factor.

${(y - 7)}^{2} = 9 \cdot 9 \frac{x^{2} - 6 x + 153}{9 \cdot 16}$

Step 4.2.1.1.4

Rewrite the expression.

${(y - 7)}^{2} = 9 \frac{x^{2} - 6 x + 153}{16}$

Step 4.2.1.2

Combine $9$ and $\frac{x^{2} - 6 x + 153}{16}$ .

${(y - 7)}^{2} = \frac{9 (x^{2} - 6 x + 153)}{16}$

Step 5

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

$y - 7 = \pm \sqrt{\frac{9 (x^{2} - 6 x + 153)}{16}}$

Step 6

Simplify $\pm \sqrt{\frac{9 (x^{2} - 6 x + 153)}{16}}$ .

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Step 6.1

Rewrite $\frac{9 (x^{2} - 6 x + 153)}{16}$ as ${(\frac{3}{4})}^{2} (x^{2} - 6 x + 153)$ .

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Step 6.1.1

Factor the perfect power $3^{2}$ out of $9 (x^{2} - 6 x + 153)$ .

$y - 7 = \pm \sqrt{\frac{3^{2} (x^{2} - 6 x + 153)}{16}}$

Step 6.1.2

Factor the perfect power $4^{2}$ out of $16$ .

$y - 7 = \pm \sqrt{\frac{3^{2} (x^{2} - 6 x + 153)}{4^{2} \cdot 1}}$

Step 6.1.3

Rearrange the fraction $\frac{3^{2} (x^{2} - 6 x + 153)}{4^{2} \cdot 1}$ .

$y - 7 = \pm \sqrt{{(\frac{3}{4})}^{2} (x^{2} - 6 x + 153)}$

Step 6.2

Pull terms out from under the radical.

$y - 7 = \pm \frac{3}{4} \sqrt{x^{2} - 6 x + 153}$

Step 6.3

Combine $\frac{3}{4}$ and $\sqrt{x^{2} - 6 x + 153}$ .

$y - 7 = \pm \frac{3 \sqrt{x^{2} - 6 x + 153}}{4}$

Step 7

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

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Step 7.1

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

$y - 7 = \frac{3 \sqrt{x^{2} - 6 x + 153}}{4}$

Step 7.2

Add $7$ to both sides of the equation.

$y = \frac{3 \sqrt{x^{2} - 6 x + 153}}{4} + 7$

Step 7.3

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

$y - 7 = - \frac{3 \sqrt{x^{2} - 6 x + 153}}{4}$

Step 7.4

Add $7$ to both sides of the equation.

$y = - \frac{3 \sqrt{x^{2} - 6 x + 153}}{4} + 7$

Step 7.5

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

$y = \frac{3 \sqrt{x^{2} - 6 x + 153}}{4} + 7$

$y = - \frac{3 \sqrt{x^{2} - 6 x + 153}}{4} + 7$

$y = \frac{3 \sqrt{x^{2} - 6 x + 153}}{4} + 7$

$y = - \frac{3 \sqrt{x^{2} - 6 x + 153}}{4} + 7$

Step 8

Set the radicand in $\sqrt{x^{2} - 6 x + 153}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} - 6 x + 153 \geq 0$

Step 9

Solve for $x$ .

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Step 9.1

Convert the inequality to an equation.

$x^{2} - 6 x + 153 = 0$

Step 9.2

Use the quadratic formula to find the solutions.

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

Step 9.3

Substitute the values $a = 1$ , $b = - 6$ , and $c = 153$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 153)}}{2 \cdot 1}$

Step 9.4

Simplify.

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Step 9.4.1

Simplify the numerator.

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Step 9.4.1.1

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 153}}{2 \cdot 1}$

Step 9.4.1.2

Multiply $- 4 \cdot 1 \cdot 153$ .

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Step 9.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 153}}{2 \cdot 1}$

Step 9.4.1.2.2

Multiply $- 4$ by $153$ .

$x = \frac{6 \pm \sqrt{36 - 612}}{2 \cdot 1}$

Step 9.4.1.3

Subtract $612$ from $36$ .

$x = \frac{6 \pm \sqrt{- 576}}{2 \cdot 1}$

Step 9.4.1.4

Rewrite $- 576$ as $- 1 (576)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 576}}{2 \cdot 1}$

Step 9.4.1.5

Rewrite $\sqrt{- 1 (576)}$ as $\sqrt{- 1} \cdot \sqrt{576}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{576}}{2 \cdot 1}$

Step 9.4.1.6

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

$x = \frac{6 \pm i \cdot \sqrt{576}}{2 \cdot 1}$

Step 9.4.1.7

Rewrite $576$ as $24^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{24^{2}}}{2 \cdot 1}$

Step 9.4.1.8

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

$x = \frac{6 \pm i \cdot 24}{2 \cdot 1}$

Step 9.4.1.9

Move $24$ to the left of $i$ .

$x = \frac{6 \pm 24 i}{2 \cdot 1}$

Step 9.4.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 24 i}{2}$

Step 9.4.3

Simplify $\frac{6 \pm 24 i}{2}$ .

$x = 3 \pm 12 i$

Step 9.5

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

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Step 9.5.1

Simplify the numerator.

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Step 9.5.1.1

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 153}}{2 \cdot 1}$

Step 9.5.1.2

Multiply $- 4 \cdot 1 \cdot 153$ .

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Step 9.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 153}}{2 \cdot 1}$

Step 9.5.1.2.2

Multiply $- 4$ by $153$ .

$x = \frac{6 \pm \sqrt{36 - 612}}{2 \cdot 1}$

Step 9.5.1.3

Subtract $612$ from $36$ .

$x = \frac{6 \pm \sqrt{- 576}}{2 \cdot 1}$

Step 9.5.1.4

Rewrite $- 576$ as $- 1 (576)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 576}}{2 \cdot 1}$

Step 9.5.1.5

Rewrite $\sqrt{- 1 (576)}$ as $\sqrt{- 1} \cdot \sqrt{576}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{576}}{2 \cdot 1}$

Step 9.5.1.6

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

$x = \frac{6 \pm i \cdot \sqrt{576}}{2 \cdot 1}$

Step 9.5.1.7

Rewrite $576$ as $24^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{24^{2}}}{2 \cdot 1}$

Step 9.5.1.8

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

$x = \frac{6 \pm i \cdot 24}{2 \cdot 1}$

Step 9.5.1.9

Move $24$ to the left of $i$ .

$x = \frac{6 \pm 24 i}{2 \cdot 1}$

Step 9.5.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 24 i}{2}$

Step 9.5.3

Simplify $\frac{6 \pm 24 i}{2}$ .

$x = 3 \pm 12 i$

Step 9.5.4

Change the $\pm$ to $+$ .

$x = 3 + 12 i$

Step 9.6

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

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Step 9.6.1

Simplify the numerator.

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Step 9.6.1.1

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 153}}{2 \cdot 1}$

Step 9.6.1.2

Multiply $- 4 \cdot 1 \cdot 153$ .

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Step 9.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 153}}{2 \cdot 1}$

Step 9.6.1.2.2

Multiply $- 4$ by $153$ .

$x = \frac{6 \pm \sqrt{36 - 612}}{2 \cdot 1}$

Step 9.6.1.3

Subtract $612$ from $36$ .

$x = \frac{6 \pm \sqrt{- 576}}{2 \cdot 1}$

Step 9.6.1.4

Rewrite $- 576$ as $- 1 (576)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 576}}{2 \cdot 1}$

Step 9.6.1.5

Rewrite $\sqrt{- 1 (576)}$ as $\sqrt{- 1} \cdot \sqrt{576}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{576}}{2 \cdot 1}$

Step 9.6.1.6

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

$x = \frac{6 \pm i \cdot \sqrt{576}}{2 \cdot 1}$

Step 9.6.1.7

Rewrite $576$ as $24^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{24^{2}}}{2 \cdot 1}$

Step 9.6.1.8

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

$x = \frac{6 \pm i \cdot 24}{2 \cdot 1}$

Step 9.6.1.9

Move $24$ to the left of $i$ .

$x = \frac{6 \pm 24 i}{2 \cdot 1}$

Step 9.6.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 24 i}{2}$

Step 9.6.3

Simplify $\frac{6 \pm 24 i}{2}$ .

$x = 3 \pm 12 i$

Step 9.6.4

Change the $\pm$ to $-$ .

$x = 3 - 12 i$

Step 9.7

Identify the leading coefficient.

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Step 9.7.1

The leading term in a polynomial is the term with the highest degree.

$x^{2}$

Step 9.7.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$1$

Step 9.8

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $x^{2} - 6 x + 153$ is always greater than $0$ .

All real numbers

Step 10

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 11

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

Interval Notation:

$(- \infty, - 2] \cup [16, \infty)$

Set-Builder Notation:

${y | y \leq - 2, y \geq 16}$

Step 12

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $(- \infty, - 2] \cup [16, \infty), {y | y \leq - 2, y \geq 16}$

Step 13