Linear Algebra Examples

$x + 5 \sqrt{x^{2} + 11 x + 3}$

Step 1

Set the radicand in $\sqrt{x^{2} + 11 x + 3}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} + 11 x + 3 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Convert the inequality to an equation.

$x^{2} + 11 x + 3 = 0$

Step 2.2

Use the quadratic formula to find the solutions.

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

Step 2.3

Substitute the values $a = 1$ , $b = 11$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{- 11 \pm \sqrt{11^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 2.4.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 3}}{2 \cdot 1}$

Step 2.4.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 11 \pm \sqrt{121 - 12}}{2 \cdot 1}$

Step 2.4.1.3

Subtract $12$ from $121$ .

$x = \frac{- 11 \pm \sqrt{109}}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 11 \pm \sqrt{109}}{2}$

Step 2.5

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

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 3}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 11 \pm \sqrt{121 - 12}}{2 \cdot 1}$

Step 2.5.1.3

Subtract $12$ from $121$ .

$x = \frac{- 11 \pm \sqrt{109}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 11 \pm \sqrt{109}}{2}$

Step 2.5.3

Change the $\pm$ to $+$ .

$x = \frac{- 11 + \sqrt{109}}{2}$

Step 2.5.4

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 1 \cdot 11 + \sqrt{109}}{2}$

Step 2.5.5

Factor $- 1$ out of $\sqrt{109}$ .

$x = \frac{- 1 \cdot 11 - 1 (- \sqrt{109})}{2}$

Step 2.5.6

Factor $- 1$ out of $- 1 (11) - 1 (- \sqrt{109})$ .

$x = \frac{- 1 (11 - \sqrt{109})}{2}$

Step 2.5.7

Move the negative in front of the fraction.

$x = - \frac{11 - \sqrt{109}}{2}$

Step 2.6

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

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $11$ to the power of $2$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 11 \pm \sqrt{121 - 4 \cdot 3}}{2 \cdot 1}$

Step 2.6.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{- 11 \pm \sqrt{121 - 12}}{2 \cdot 1}$

Step 2.6.1.3

Subtract $12$ from $121$ .

$x = \frac{- 11 \pm \sqrt{109}}{2 \cdot 1}$

Step 2.6.2

Multiply $2$ by $1$ .

$x = \frac{- 11 \pm \sqrt{109}}{2}$

Step 2.6.3

Change the $\pm$ to $-$ .

$x = \frac{- 11 - \sqrt{109}}{2}$

Step 2.6.4

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 1 \cdot 11 - \sqrt{109}}{2}$

Step 2.6.5

Factor $- 1$ out of $- \sqrt{109}$ .

$x = \frac{- 1 \cdot 11 - (\sqrt{109})}{2}$

Step 2.6.6

Factor $- 1$ out of $- 1 (11) - (\sqrt{109})$ .

$x = \frac{- 1 (11 + \sqrt{109})}{2}$

Step 2.6.7

Move the negative in front of the fraction.

$x = - \frac{11 + \sqrt{109}}{2}$

Step 2.7

Consolidate the solutions.

$x = - \frac{11 - \sqrt{109}}{2}, - \frac{11 + \sqrt{109}}{2}$

Step 2.8

Use each root to create test intervals.

$x < - \frac{11 + \sqrt{109}}{2}$

$- \frac{11 + \sqrt{109}}{2} < x < - \frac{11 - \sqrt{109}}{2}$

$x > - \frac{11 - \sqrt{109}}{2}$

Step 2.9

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 2.9.1

Test a value on the interval $x < - \frac{11 + \sqrt{109}}{2}$ to see if it makes the inequality true.

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Step 2.9.1.1

Choose a value on the interval $x < - \frac{11 + \sqrt{109}}{2}$ and see if this value makes the original inequality true.

$x = - 13$

Step 2.9.1.2

Replace $x$ with $- 13$ in the original inequality.

${(- 13)}^{2} + 11 (- 13) + 3 \geq 0$

Step 2.9.1.3

The left side $29$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.9.2

Test a value on the interval $- \frac{11 + \sqrt{109}}{2} < x < - \frac{11 - \sqrt{109}}{2}$ to see if it makes the inequality true.

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Step 2.9.2.1

Choose a value on the interval $- \frac{11 + \sqrt{109}}{2} < x < - \frac{11 - \sqrt{109}}{2}$ and see if this value makes the original inequality true.

$x = - 3$

Step 2.9.2.2

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

${(- 3)}^{2} + 11 (- 3) + 3 \geq 0$

Step 2.9.2.3

The left side $- 21$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.9.3

Test a value on the interval $x > - \frac{11 - \sqrt{109}}{2}$ to see if it makes the inequality true.

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Step 2.9.3.1

Choose a value on the interval $x > - \frac{11 - \sqrt{109}}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 2.9.3.2

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

${(0)}^{2} + 11 (0) + 3 \geq 0$

Step 2.9.3.3

The left side $3$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.9.4

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

$x < - \frac{11 + \sqrt{109}}{2}$ True

$- \frac{11 + \sqrt{109}}{2} < x < - \frac{11 - \sqrt{109}}{2}$ False

$x > - \frac{11 - \sqrt{109}}{2}$ True

$x < - \frac{11 + \sqrt{109}}{2}$ True

$- \frac{11 + \sqrt{109}}{2} < x < - \frac{11 - \sqrt{109}}{2}$ False

$x > - \frac{11 - \sqrt{109}}{2}$ True

Step 2.10

The solution consists of all of the true intervals.

$x \leq - \frac{11 + \sqrt{109}}{2}$ or $x \geq - \frac{11 - \sqrt{109}}{2}$

Step 3

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

Interval Notation:

$(- \infty, - \frac{11 + \sqrt{109}}{2}] \cup [- \frac{11 - \sqrt{109}}{2}, \infty)$

Set-Builder Notation:

${x | x \leq - \frac{11 + \sqrt{109}}{2}, x \geq - \frac{11 - \sqrt{109}}{2}}$

Step 4