Algebra Examples

$\sqrt{x^{2} - 14 x + 49} = x - 7$

Step 1

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

${\sqrt{x^{2} - 14 x + 49}}^{2} = {(x - 7)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 14 x + 49}$ as ${(x^{2} - 14 x + 49)}^{\frac{1}{2}}$ .

${({(x^{2} - 14 x + 49)}^{\frac{1}{2}})}^{2} = {(x - 7)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x^{2} - 14 x + 49)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x^{2} - 14 x + 49)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(x^{2} - 14 x + 49)}^{\frac{1}{2} \cdot 2} = {(x - 7)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x^{2} - 14 x + 49)}^{\frac{1}{2} \cdot 2} = {(x - 7)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x^{2} - 14 x + 49)}^{1} = {(x - 7)}^{2}$

Step 2.2.1.2

Simplify.

$x^{2} - 14 x + 49 = {(x - 7)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(x - 7)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(x - 7)}^{2}$ as $(x - 7) (x - 7)$ .

$x^{2} - 14 x + 49 = (x - 7) (x - 7)$

Step 2.3.1.2

Expand $(x - 7) (x - 7)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$x^{2} - 14 x + 49 = x (x - 7) - 7 (x - 7)$

Step 2.3.1.2.2

Apply the distributive property.

$x^{2} - 14 x + 49 = x \cdot x + x \cdot - 7 - 7 (x - 7)$

Step 2.3.1.2.3

Apply the distributive property.

$x^{2} - 14 x + 49 = x \cdot x + x \cdot - 7 - 7 x - 7 \cdot - 7$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} - 14 x + 49 = x^{2} + x \cdot - 7 - 7 x - 7 \cdot - 7$

Step 2.3.1.3.1.2

Move $- 7$ to the left of $x$ .

$x^{2} - 14 x + 49 = x^{2} - 7 \cdot x - 7 x - 7 \cdot - 7$

Step 2.3.1.3.1.3

Multiply $- 7$ by $- 7$ .

$x^{2} - 14 x + 49 = x^{2} - 7 x - 7 x + 49$

Step 2.3.1.3.2

Subtract $7 x$ from $- 7 x$ .

$x^{2} - 14 x + 49 = x^{2} - 14 x + 49$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} - 14 x + 49 - x^{2} = - 14 x + 49$

Step 3.1.2

Add $14 x$ to both sides of the equation.

$x^{2} - 14 x + 49 - x^{2} + 14 x = 49$

Step 3.1.3

Combine the opposite terms in $x^{2} - 14 x + 49 - x^{2} + 14 x$ .

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Step 3.1.3.1

Subtract $x^{2}$ from $x^{2}$ .

$- 14 x + 49 + 0 + 14 x = 49$

Step 3.1.3.2

Add $- 14 x + 49$ and $0$ .

$- 14 x + 49 + 14 x = 49$

Step 3.1.3.3

Add $- 14 x$ and $14 x$ .

$0 + 49 = 49$

Step 3.1.3.4

Add $0$ and $49$ .

$49 = 49$

Step 3.2

Since $49 = 49$ , the equation will always be true.

Always true

Step 4

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$