Solve the Inequality for x square root of x^2-4x+4>0

$\sqrt{x^{2} - 4 x + 4} > 0$

Step 1

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

${\sqrt{x^{2} - 4 x + 4}}^{2} > 0^{2}$

Step 2

Simplify each side of the inequality.

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

${({(x^{2} - 4 x + 4)}^{\frac{1}{2}})}^{2} > 0^{2}$

Simplify the left side.

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

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

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

${(x^{2} - 4 x + 4)}^{\frac{1}{2} \cdot 2} > 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} - 4 x + 4)}^{\frac{1}{2} \cdot 2} > 0^{2}$

Rewrite the expression.

${(x^{2} - 4 x + 4)}^{1} > 0^{2}$

Simplify.

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

Simplify the right side.

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

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

Step 3

Solve for $x$ .

Convert the inequality to an equation.

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

Factor using the perfect square rule.

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

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

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

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

Rewrite the polynomial.

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

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

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 4

Use each root to create test intervals.

$x < 2$

$x > 2$

Step 5

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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

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

$x = 0$

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

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

The left side $2$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = 4$

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

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

The left side $2$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

$x < 2$ True

$x > 2$ True

$x < 2$ True

$x > 2$ True

Step 6

The solution consists of all of the true intervals.

$x < 2$ or $x > 2$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x < 2 or x > 2$

Interval Notation:

$(- \infty, 2) \cup (2, \infty)$

Step 8

Solve the Inequality for x square root of x^2-4x+4>0

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