Finite Math Examples

${(x + \frac{3}{4})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{25}{16}$

Step 1

Subtract ${(x + \frac{3}{4})}^{2}$ from both sides of the equation.

${(y - \frac{1}{2})}^{2} = \frac{25}{16} - {(x + \frac{3}{4})}^{2}$

Step 2

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

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - {(x + \frac{3}{4})}^{2}}$

Step 3

Simplify $\pm \sqrt{\frac{25}{16} - {(x + \frac{3}{4})}^{2}}$ .

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Step 3.1

Rewrite ${(x + \frac{3}{4})}^{2}$ as $(x + \frac{3}{4}) (x + \frac{3}{4})$ .

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - ((x + \frac{3}{4}) (x + \frac{3}{4}))}$

Step 3.2

Expand $(x + \frac{3}{4}) (x + \frac{3}{4})$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x (x + \frac{3}{4}) + \frac{3}{4} (x + \frac{3}{4}))}$

Step 3.2.2

Apply the distributive property.

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x \cdot x + x \frac{3}{4} + \frac{3}{4} (x + \frac{3}{4}))}$

Step 3.2.3

Apply the distributive property.

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x \cdot x + x \frac{3}{4} + \frac{3}{4} x + \frac{3}{4} \cdot \frac{3}{4})}$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $x$ by $x$ .

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x^{2} + x \frac{3}{4} + \frac{3}{4} x + \frac{3}{4} \cdot \frac{3}{4})}$

Step 3.3.1.2

Combine $x$ and $\frac{3}{4}$ .

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x^{2} + \frac{x \cdot 3}{4} + \frac{3}{4} x + \frac{3}{4} \cdot \frac{3}{4})}$

Step 3.3.1.3

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

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x^{2} + \frac{3 \cdot x}{4} + \frac{3}{4} x + \frac{3}{4} \cdot \frac{3}{4})}$

Step 3.3.1.4

Combine $\frac{3}{4}$ and $x$ .

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x^{2} + \frac{3 x}{4} + \frac{3 x}{4} + \frac{3}{4} \cdot \frac{3}{4})}$

Step 3.3.1.5

Multiply $\frac{3}{4} \cdot \frac{3}{4}$ .

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Step 3.3.1.5.1

Multiply $\frac{3}{4}$ by $\frac{3}{4}$ .

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x^{2} + \frac{3 x}{4} + \frac{3 x}{4} + \frac{3 \cdot 3}{4 \cdot 4})}$

Step 3.3.1.5.2

Multiply $3$ by $3$ .

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x^{2} + \frac{3 x}{4} + \frac{3 x}{4} + \frac{9}{4 \cdot 4})}$

Step 3.3.1.5.3

Multiply $4$ by $4$ .

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x^{2} + \frac{3 x}{4} + \frac{3 x}{4} + \frac{9}{16})}$

Step 3.3.2

Add $\frac{3 x}{4}$ and $\frac{3 x}{4}$ .

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x^{2} + 2 \frac{3 x}{4} + \frac{9}{16})}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Factor $2$ out of $4$ .

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x^{2} + 2 \frac{3 x}{2 (2)} + \frac{9}{16})}$

Step 3.4.2

Cancel the common factor.

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x^{2} + 2 \frac{3 x}{2 \cdot 2} + \frac{9}{16})}$

Step 3.4.3

Rewrite the expression.

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - (x^{2} + \frac{3 x}{2} + \frac{9}{16})}$

Step 3.5

Apply the distributive property.

$y - \frac{1}{2} = \pm \sqrt{\frac{25}{16} - x^{2} - \frac{3 x}{2} - \frac{9}{16}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Combine the numerators over the common denominator.

$y - \frac{1}{2} = \pm \sqrt{- x^{2} - \frac{3 x}{2} + \frac{25 - 9}{16}}$

Step 3.6.2

Subtract $9$ from $25$ .

$y - \frac{1}{2} = \pm \sqrt{- x^{2} - \frac{3 x}{2} + \frac{16}{16}}$

Step 3.6.3

Divide $16$ by $16$ .

$y - \frac{1}{2} = \pm \sqrt{- x^{2} - \frac{3 x}{2} + 1}$

Step 3.7

To write $- x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.8

Simplify terms.

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Step 3.8.1

Combine $- x^{2}$ and $\frac{2}{2}$ .

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

Step 3.8.2

Combine the numerators over the common denominator.

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

Step 3.9

Simplify the numerator.

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Step 3.9.1

Factor $x$ out of $- x^{2} \cdot 2 - 3 x$ .

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Step 3.9.1.1

Factor $x$ out of $- x^{2} \cdot 2$ .

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

Step 3.9.1.2

Factor $x$ out of $- 3 x$ .

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

Step 3.9.1.3

Factor $x$ out of $x (- x \cdot 2) + x \cdot - 3$ .

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

Step 3.9.2

Multiply $2$ by $- 1$ .

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

Step 3.10

Combine into one fraction.

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Step 3.10.1

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

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

Step 3.10.2

Combine the numerators over the common denominator.

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

Step 3.11

Simplify the numerator.

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Step 3.11.1

Apply the distributive property.

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

Step 3.11.2

Rewrite using the commutative property of multiplication.

$y - \frac{1}{2} = \pm \sqrt{\frac{- 2 x \cdot x + x \cdot - 3 + 2}{2}}$

Step 3.11.3

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

$y - \frac{1}{2} = \pm \sqrt{\frac{- 2 x \cdot x - 3 \cdot x + 2}{2}}$

Step 3.11.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.11.4.1

Move $x$ .

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

Step 3.11.4.2

Multiply $x$ by $x$ .

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

$y - \frac{1}{2} = \pm \sqrt{\frac{- 2 x^{2} - 3 x + 2}{2}}$

Step 3.11.5

Factor by grouping.

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Step 3.11.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 2 \cdot 2 = - 4$ and whose sum is $b = - 3$ .

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Step 3.11.5.1.1

Factor $- 3$ out of $- 3 x$ .

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

Step 3.11.5.1.2

Rewrite $- 3$ as $1$ plus $- 4$

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

Step 3.11.5.1.3

Apply the distributive property.

$y - \frac{1}{2} = \pm \sqrt{\frac{- 2 x^{2} + 1 x - 4 x + 2}{2}}$

Step 3.11.5.1.4

Multiply $x$ by $1$ .

$y - \frac{1}{2} = \pm \sqrt{\frac{- 2 x^{2} + x - 4 x + 2}{2}}$

Step 3.11.5.2

Factor out the greatest common factor from each group.

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Step 3.11.5.2.1

Group the first two terms and the last two terms.

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

Step 3.11.5.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.11.5.3

Factor the polynomial by factoring out the greatest common factor, $- 2 x + 1$ .

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

Step 3.12

Rewrite $\sqrt{\frac{(- 2 x + 1) (x + 2)}{2}}$ as $\frac{\sqrt{(- 2 x + 1) (x + 2)}}{\sqrt{2}}$ .

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

Step 3.13

Multiply $\frac{\sqrt{(- 2 x + 1) (x + 2)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.14

Combine and simplify the denominator.

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Step 3.14.1

Multiply $\frac{\sqrt{(- 2 x + 1) (x + 2)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.14.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.14.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.14.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.14.5

Add $1$ and $1$ .

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

Step 3.14.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.14.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 3.14.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.14.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.14.6.4

Cancel the common factor of $2$ .

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Step 3.14.6.4.1

Cancel the common factor.

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

Step 3.14.6.4.2

Rewrite the expression.

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

Step 3.14.6.5

Evaluate the exponent.

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

Step 3.15

Combine using the product rule for radicals.

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

Step 3.16

Reorder factors in $\pm \frac{\sqrt{(- 2 x + 1) (x + 2) \cdot 2}}{2}$ .

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

Step 4

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

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Step 4.1

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

$y - \frac{1}{2} = \frac{\sqrt{2 (- 2 x + 1) (x + 2)}}{2}$

Step 4.2

Add $\frac{1}{2}$ to both sides of the equation.

$y = \frac{\sqrt{2 (- 2 x + 1) (x + 2)}}{2} + \frac{1}{2}$

Step 4.3

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

$y - \frac{1}{2} = - \frac{\sqrt{2 (- 2 x + 1) (x + 2)}}{2}$

Step 4.4

Add $\frac{1}{2}$ to both sides of the equation.

$y = - \frac{\sqrt{2 (- 2 x + 1) (x + 2)}}{2} + \frac{1}{2}$

Step 4.5

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

$y = \frac{\sqrt{2 (- 2 x + 1) (x + 2)}}{2} + \frac{1}{2}$

$y = - \frac{\sqrt{2 (- 2 x + 1) (x + 2)}}{2} + \frac{1}{2}$

$y = \frac{\sqrt{2 (- 2 x + 1) (x + 2)}}{2} + \frac{1}{2}$

$y = - \frac{\sqrt{2 (- 2 x + 1) (x + 2)}}{2} + \frac{1}{2}$

Step 5

Set the radicand in $\sqrt{2 (- 2 x + 1) (x + 2)}$ greater than or equal to $0$ to find where the expression is defined.

$2 (- 2 x + 1) (x + 2) \geq 0$

Step 6

Solve for $x$ .

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Step 6.1

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

$- 2 x + 1 = 0$

$x + 2 = 0$

Step 6.2

Set $- 2 x + 1$ equal to $0$ and solve for $x$ .

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Step 6.2.1

Set $- 2 x + 1$ equal to $0$ .

$- 2 x + 1 = 0$

Step 6.2.2

Solve $- 2 x + 1 = 0$ for $x$ .

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Step 6.2.2.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 6.2.2.2

Divide each term in $- 2 x = - 1$ by $- 2$ and simplify.

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Step 6.2.2.2.1

Divide each term in $- 2 x = - 1$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Step 6.2.2.2.2

Simplify the left side.

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Step 6.2.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 6.2.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Step 6.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 2}$

Step 6.2.2.2.3

Simplify the right side.

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Step 6.2.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{2}$

Step 6.3

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 6.3.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 6.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6.4

The final solution is all the values that make $2 (- 2 x + 1) (x + 2) \geq 0$ true.

$x = \frac{1}{2}, - 2$

Step 6.5

Use each root to create test intervals.

$x < - 2$

$- 2 < x < \frac{1}{2}$

$x > \frac{1}{2}$

Step 6.6

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 6.6.1

Test a value on the interval $x < - 2$ to see if it makes the inequality true.

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Step 6.6.1.1

Choose a value on the interval $x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Step 6.6.1.2

Replace $x$ with $- 4$ in the original inequality.

$2 (- 2 \cdot - 4 + 1) ((- 4) + 2) \geq 0$

Step 6.6.1.3

The left side $- 36$ is less than the right side $0$ , which means that the given statement is false.

False

Step 6.6.2

Test a value on the interval $- 2 < x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 6.6.2.1

Choose a value on the interval $- 2 < x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 6.6.2.2

Replace $x$ with $0$ in the original inequality.

$2 (- 2 \cdot 0 + 1) ((0) + 2) \geq 0$

Step 6.6.2.3

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

True

Step 6.6.3

Test a value on the interval $x > \frac{1}{2}$ to see if it makes the inequality true.

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Step 6.6.3.1

Choose a value on the interval $x > \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 3$

Step 6.6.3.2

Replace $x$ with $3$ in the original inequality.

$2 (- 2 \cdot 3 + 1) ((3) + 2) \geq 0$

Step 6.6.3.3

The left side $- 50$ is less than the right side $0$ , which means that the given statement is false.

False

Step 6.6.4

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

$x < - 2$ False

$- 2 < x < \frac{1}{2}$ True

$x > \frac{1}{2}$ False

$x < - 2$ False

$- 2 < x < \frac{1}{2}$ True

$x > \frac{1}{2}$ False

Step 6.7

The solution consists of all of the true intervals.

$- 2 \leq x \leq \frac{1}{2}$

Step 7

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

Interval Notation:

$[- 2, \frac{1}{2}]$

Set-Builder Notation:

${x | - 2 \leq x \leq \frac{1}{2}}$

Step 8

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

Interval Notation:

$[- \frac{3}{4}, \frac{7}{4}]$

Set-Builder Notation:

${y | - \frac{3}{4} \leq y \leq \frac{7}{4}}$

Step 9

Determine the domain and range.

Domain: $[- 2, \frac{1}{2}], {x | - 2 \leq x \leq \frac{1}{2}}$

Range: $[- \frac{3}{4}, \frac{7}{4}], {y | - \frac{3}{4} \leq y \leq \frac{7}{4}}$

Step 10