Algebra Examples

$\sqrt{x^{2} - 4 x + 4}$

Step 1

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

$x^{2} - 4 x + 4 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Convert the inequality to an equation.

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

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

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

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

Step 2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 2.2.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 2 + 2^{2} = 0$

Step 2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

${(x - 2)}^{2} = 0$

Step 2.3

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

$x - 2 = 0$

Step 2.4

Add $2$ to both sides of the equation.

$x = 2$

Step 2.5

Use each root to create test intervals.

$x < 2$

$x > 2$

Step 2.6

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 2.6.1

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

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Step 2.6.1.1

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

$x = 0$

Step 2.6.1.2

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

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

Step 2.6.1.3

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

True

Step 2.6.2

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

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Step 2.6.2.1

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

$x = 4$

Step 2.6.2.2

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

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

Step 2.6.2.3

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

True

Step 2.6.3

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

$x < 2$ True

$x > 2$ True

$x < 2$ True

$x > 2$ True

Step 2.7

The solution consists of all of the true intervals.

$x \leq 2$ or $x \geq 2$

Step 2.8

Combine the intervals.

All real numbers

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

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

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 6