Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

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

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

Step 1.1.2

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 1.1.3

Rewrite the polynomial.

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

Step 1.1.4

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

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

Step 1.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 1.3

Factor using the perfect square rule.

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Step 1.3.1

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

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

Step 1.3.2

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 1.3.3

Rewrite the polynomial.

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

Step 1.3.4

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

$x - 2 - \sqrt{{(x + 2)}^{2}}$

Step 1.4

Pull terms out from under the radical, assuming positive real numbers.

$x - 2 - (x + 2)$

Step 1.5

Apply the distributive property.

$x - 2 - x - 1 \cdot 2$

Step 1.6

Multiply $- 1$ by $2$ .

$x - 2 - x - 2$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $x - 2 - x - 2$ .

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Step 2.1.1

Subtract $x$ from $x$ .

$0 - 2 - 2$

Step 2.1.2

Subtract $2$ from $0$ .

$- 2 - 2$

Step 2.2

Subtract $2$ from $- 2$ .

$- 4$