Precalculus Examples

$y = \frac{4}{\sqrt{4 - 8 x - 5 x^{2}}}$

Step 1

Set the radicand in $\sqrt{4 - 8 x - 5 x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$4 - 8 x - 5 x^{2} \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Convert the inequality to an equation.

$4 - 8 x - 5 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $- 1$ out of $4 - 8 x - 5 x^{2}$ .

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Step 2.2.1.1

Reorder the expression.

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Step 2.2.1.1.1

Move $4$ .

$- 8 x - 5 x^{2} + 4 = 0$

Step 2.2.1.1.2

Reorder $- 8 x$ and $- 5 x^{2}$ .

$- 5 x^{2} - 8 x + 4 = 0$

Step 2.2.1.2

Factor $- 1$ out of $- 5 x^{2}$ .

$- (5 x^{2}) - 8 x + 4 = 0$

Step 2.2.1.3

Factor $- 1$ out of $- 8 x$ .

$- (5 x^{2}) - (8 x) + 4 = 0$

Step 2.2.1.4

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

$- (5 x^{2}) - (8 x) - 1 \cdot - 4 = 0$

Step 2.2.1.5

Factor $- 1$ out of $- (5 x^{2}) - (8 x)$ .

$- (5 x^{2} + 8 x) - 1 \cdot - 4 = 0$

Step 2.2.1.6

Factor $- 1$ out of $- (5 x^{2} + 8 x) - 1 (- 4)$ .

$- (5 x^{2} + 8 x - 4) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor by grouping.

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Step 2.2.2.1.1

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 $a \cdot c = 5 \cdot - 4 = - 20$ and whose sum is $b = 8$ .

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Step 2.2.2.1.1.1

Factor $8$ out of $8 x$ .

$- (5 x^{2} + 8 (x) - 4) = 0$

Step 2.2.2.1.1.2

Rewrite $8$ as $- 2$ plus $10$

$- (5 x^{2} + (- 2 + 10) x - 4) = 0$

Step 2.2.2.1.1.3

Apply the distributive property.

$- (5 x^{2} - 2 x + 10 x - 4) = 0$

Step 2.2.2.1.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1.2.1

Group the first two terms and the last two terms.

$- ((5 x^{2} - 2 x) + 10 x - 4) = 0$

Step 2.2.2.1.2.2

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

$- (x (5 x - 2) + 2 (5 x - 2)) = 0$

Step 2.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 x - 2$ .

$- ((5 x - 2) (x + 2)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$- (5 x - 2) (x + 2) = 0$

Step 2.3

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

$5 x - 2 = 0$

$x + 2 = 0$

Step 2.4

Set $5 x - 2$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $5 x - 2$ equal to $0$ .

$5 x - 2 = 0$

Step 2.4.2

Solve $5 x - 2 = 0$ for $x$ .

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Step 2.4.2.1

Add $2$ to both sides of the equation.

$5 x = 2$

Step 2.4.2.2

Divide each term in $5 x = 2$ by $5$ and simplify.

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Step 2.4.2.2.1

Divide each term in $5 x = 2$ by $5$ .

$\frac{5 x}{5} = \frac{2}{5}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{2}{5}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{5}$

Step 2.5

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

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Step 2.5.1

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

$x + 2 = 0$

Step 2.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.6

The final solution is all the values that make $- (5 x - 2) (x + 2) = 0$ true.

$x = \frac{2}{5}, - 2$

Step 2.7

Use each root to create test intervals.

$x < - 2$

$- 2 < x < \frac{2}{5}$

$x > \frac{2}{5}$

Step 2.8

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 2.8.1

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

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Step 2.8.1.1

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

$x = - 4$

Step 2.8.1.2

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

$4 - 8 \cdot - 4 - 5 {(- 4)}^{2} \geq 0$

Step 2.8.1.3

The left side $- 44$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.8.2

Test a value on the interval $- 2 < x < \frac{2}{5}$ to see if it makes the inequality true.

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Step 2.8.2.1

Choose a value on the interval $- 2 < x < \frac{2}{5}$ and see if this value makes the original inequality true.

$x = 0$

Step 2.8.2.2

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

$4 - 8 \cdot 0 - 5 {(0)}^{2} \geq 0$

Step 2.8.2.3

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

True

Step 2.8.3

Test a value on the interval $x > \frac{2}{5}$ to see if it makes the inequality true.

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Step 2.8.3.1

Choose a value on the interval $x > \frac{2}{5}$ and see if this value makes the original inequality true.

$x = 3$

Step 2.8.3.2

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

$4 - 8 \cdot 3 - 5 {(3)}^{2} \geq 0$

Step 2.8.3.3

The left side $- 65$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.8.4

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

$x < - 2$ False

$- 2 < x < \frac{2}{5}$ True

$x > \frac{2}{5}$ False

$x < - 2$ False

$- 2 < x < \frac{2}{5}$ True

$x > \frac{2}{5}$ False

Step 2.9

The solution consists of all of the true intervals.

$- 2 \leq x \leq \frac{2}{5}$

Step 3

Set the denominator in $\frac{4}{\sqrt{4 - 8 x - 5 x^{2}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{4 - 8 x - 5 x^{2}} = 0$

Step 4

Solve for $x$ .

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Step 4.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{4 - 8 x - 5 x^{2}}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - 8 x - 5 x^{2}}$ as ${(4 - 8 x - 5 x^{2})}^{\frac{1}{2}}$ .

${({(4 - 8 x - 5 x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(4 - 8 x - 5 x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(4 - 8 x - 5 x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

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

${(4 - 8 x - 5 x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(4 - 8 x - 5 x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(4 - 8 x - 5 x^{2})}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$4 - 8 x - 5 x^{2} = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raising $0$ to any positive power yields $0$ .

$4 - 8 x - 5 x^{2} = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Factor the left side of the equation.

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Step 4.3.1.1

Factor $- 1$ out of $4 - 8 x - 5 x^{2}$ .

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Step 4.3.1.1.1

Reorder the expression.

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Step 4.3.1.1.1.1

Move $4$ .

$- 8 x - 5 x^{2} + 4 = 0$

Step 4.3.1.1.1.2

Reorder $- 8 x$ and $- 5 x^{2}$ .

$- 5 x^{2} - 8 x + 4 = 0$

Step 4.3.1.1.2

Factor $- 1$ out of $- 5 x^{2}$ .

$- (5 x^{2}) - 8 x + 4 = 0$

Step 4.3.1.1.3

Factor $- 1$ out of $- 8 x$ .

$- (5 x^{2}) - (8 x) + 4 = 0$

Step 4.3.1.1.4

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

$- (5 x^{2}) - (8 x) - 1 \cdot - 4 = 0$

Step 4.3.1.1.5

Factor $- 1$ out of $- (5 x^{2}) - (8 x)$ .

$- (5 x^{2} + 8 x) - 1 \cdot - 4 = 0$

Step 4.3.1.1.6

Factor $- 1$ out of $- (5 x^{2} + 8 x) - 1 (- 4)$ .

$- (5 x^{2} + 8 x - 4) = 0$

Step 4.3.1.2

Factor.

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Step 4.3.1.2.1

Factor by grouping.

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Step 4.3.1.2.1.1

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 $a \cdot c = 5 \cdot - 4 = - 20$ and whose sum is $b = 8$ .

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Step 4.3.1.2.1.1.1

Factor $8$ out of $8 x$ .

$- (5 x^{2} + 8 (x) - 4) = 0$

Step 4.3.1.2.1.1.2

Rewrite $8$ as $- 2$ plus $10$

$- (5 x^{2} + (- 2 + 10) x - 4) = 0$

Step 4.3.1.2.1.1.3

Apply the distributive property.

$- (5 x^{2} - 2 x + 10 x - 4) = 0$

Step 4.3.1.2.1.2

Factor out the greatest common factor from each group.

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Step 4.3.1.2.1.2.1

Group the first two terms and the last two terms.

$- ((5 x^{2} - 2 x) + 10 x - 4) = 0$

Step 4.3.1.2.1.2.2

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

$- (x (5 x - 2) + 2 (5 x - 2)) = 0$

Step 4.3.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 x - 2$ .

$- ((5 x - 2) (x + 2)) = 0$

Step 4.3.1.2.2

Remove unnecessary parentheses.

$- (5 x - 2) (x + 2) = 0$

Step 4.3.2

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

$5 x - 2 = 0$

$x + 2 = 0$

Step 4.3.3

Set $5 x - 2$ equal to $0$ and solve for $x$ .

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Step 4.3.3.1

Set $5 x - 2$ equal to $0$ .

$5 x - 2 = 0$

Step 4.3.3.2

Solve $5 x - 2 = 0$ for $x$ .

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Step 4.3.3.2.1

Add $2$ to both sides of the equation.

$5 x = 2$

Step 4.3.3.2.2

Divide each term in $5 x = 2$ by $5$ and simplify.

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Step 4.3.3.2.2.1

Divide each term in $5 x = 2$ by $5$ .

$\frac{5 x}{5} = \frac{2}{5}$

Step 4.3.3.2.2.2

Simplify the left side.

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Step 4.3.3.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{2}{5}$

Step 4.3.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{5}$

Step 4.3.4

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

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Step 4.3.4.1

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

$x + 2 = 0$

Step 4.3.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.3.5

The final solution is all the values that make $- (5 x - 2) (x + 2) = 0$ true.

$x = \frac{2}{5}, - 2$

Step 5

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

Interval Notation:

$(- 2, \frac{2}{5})$

Set-Builder Notation:

${x | - 2 < x < \frac{2}{5}}$

Step 6