Algebra Examples

$\sqrt{2 x + 4} = \frac{1}{2} x + 1$

Step 1

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $2$ .

$\sqrt{2 x + 4} \cdot \frac{2}{2} = \frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2}$

Step 2

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $2$ .

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

Step 3

Simplify.

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Step 3.1

Factor $2$ out of $2 x + 4$ .

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Step 3.1.1

Factor $2$ out of $2 x$ .

$\frac{\sqrt{2 (x) + 4} \cdot 2}{2} = \frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2}$

Step 3.1.2

Factor $2$ out of $4$ .

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

Step 3.1.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

$\frac{\sqrt{2 (x + 2)} \cdot 2}{2} = \frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2}$

Step 3.2

Move $2$ to the left of $\sqrt{2 (x + 2)}$ .

$\frac{2 \sqrt{2 (x + 2)}}{2} = \frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2}$

Step 4

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $2$ .

$x \cdot 2$

Step 5

Move $2$ to the left of $x$ .

$\frac{2 \sqrt{2 (x + 2)}}{2} = \frac{1}{2} \cdot \frac{2 x}{2} + 1 \cdot \frac{2}{2}$

Step 6

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $2$ .

$1 \cdot 2$

Step 7

Multiply $2$ by $1$ .

$\frac{2 \sqrt{2 (x + 2)}}{2} = \frac{1}{2} \cdot \frac{2 x}{2} + \frac{2}{2}$

Step 8

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

${(\sqrt{2 x + 4} \cdot \frac{2}{2})}^{2} = {(\frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2})}^{2}$

Step 9

Simplify each side of the equation.

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Step 9.1

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

${({(2 x + 4)}^{\frac{1}{2}} \cdot \frac{2}{2})}^{2} = {(\frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2})}^{2}$

Step 9.2

Simplify the left side.

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Step 9.2.1

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

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Step 9.2.1.1

Divide $2$ by $2$ .

${({(2 x + 4)}^{\frac{1}{2}} \cdot 1)}^{2} = {(\frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2})}^{2}$

Step 9.2.1.2

Multiply ${(2 x + 4)}^{\frac{1}{2}}$ by $1$ .

${({(2 x + 4)}^{\frac{1}{2}})}^{2} = {(\frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2})}^{2}$

Step 9.2.1.3

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

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Step 9.2.1.3.1

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

${(2 x + 4)}^{\frac{1}{2} \cdot 2} = {(\frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2})}^{2}$

Step 9.2.1.3.2

Cancel the common factor of $2$ .

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Step 9.2.1.3.2.1

Cancel the common factor.

${(2 x + 4)}^{\frac{1}{2} \cdot 2} = {(\frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2})}^{2}$

Step 9.2.1.3.2.2

Rewrite the expression.

${(2 x + 4)}^{1} = {(\frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2})}^{2}$

Step 9.2.1.4

Simplify.

$2 x + 4 = {(\frac{1}{2} x \cdot \frac{2}{2} + 1 \cdot \frac{2}{2})}^{2}$

Step 9.3

Simplify the right side.

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Step 9.3.1

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

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Step 9.3.1.1

Simplify terms.

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Step 9.3.1.1.1

Simplify each term.

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Step 9.3.1.1.1.1

Combine $\frac{1}{2}$ and $x$ .

$2 x + 4 = {(\frac{x}{2} \cdot \frac{2}{2} + 1 \cdot \frac{2}{2})}^{2}$

Step 9.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 9.3.1.1.1.2.1

Cancel the common factor.

$2 x + 4 = {(\frac{x}{2} \cdot \frac{2}{2} + 1 \cdot \frac{2}{2})}^{2}$

Step 9.3.1.1.1.2.2

Rewrite the expression.

$2 x + 4 = {(\frac{x}{2} \cdot 1 + 1 \cdot \frac{2}{2})}^{2}$

Step 9.3.1.1.1.3

Multiply $\frac{x}{2}$ by $1$ .

$2 x + 4 = {(\frac{x}{2} + 1 \cdot \frac{2}{2})}^{2}$

Step 9.3.1.1.1.4

Multiply $\frac{2}{2}$ by $1$ .

$2 x + 4 = {(\frac{x}{2} + \frac{2}{2})}^{2}$

Step 9.3.1.1.1.5

Divide $2$ by $2$ .

$2 x + 4 = {(\frac{x}{2} + 1)}^{2}$

Step 9.3.1.1.2

Rewrite ${(\frac{x}{2} + 1)}^{2}$ as $(\frac{x}{2} + 1) (\frac{x}{2} + 1)$ .

$2 x + 4 = (\frac{x}{2} + 1) (\frac{x}{2} + 1)$

Step 9.3.1.2

Expand $(\frac{x}{2} + 1) (\frac{x}{2} + 1)$ using the FOIL Method.

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Step 9.3.1.2.1

Apply the distributive property.

$2 x + 4 = \frac{x}{2} (\frac{x}{2} + 1) + 1 (\frac{x}{2} + 1)$

Step 9.3.1.2.2

Apply the distributive property.

$2 x + 4 = \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot 1 + 1 (\frac{x}{2} + 1)$

Step 9.3.1.2.3

Apply the distributive property.

$2 x + 4 = \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot 1 + 1 \frac{x}{2} + 1 \cdot 1$

Step 9.3.1.3

Simplify and combine like terms.

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Step 9.3.1.3.1

Simplify each term.

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Step 9.3.1.3.1.1

Multiply $\frac{x}{2} \cdot \frac{x}{2}$ .

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Step 9.3.1.3.1.1.1

Multiply $\frac{x}{2}$ by $\frac{x}{2}$ .

$2 x + 4 = \frac{x \cdot x}{2 \cdot 2} + \frac{x}{2} \cdot 1 + 1 \frac{x}{2} + 1 \cdot 1$

Step 9.3.1.3.1.1.2

Raise $x$ to the power of $1$ .

$2 x + 4 = \frac{x^{1} x}{2 \cdot 2} + \frac{x}{2} \cdot 1 + 1 \frac{x}{2} + 1 \cdot 1$

Step 9.3.1.3.1.1.3

Raise $x$ to the power of $1$ .

$2 x + 4 = \frac{x^{1} x^{1}}{2 \cdot 2} + \frac{x}{2} \cdot 1 + 1 \frac{x}{2} + 1 \cdot 1$

Step 9.3.1.3.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x + 4 = \frac{x^{1 + 1}}{2 \cdot 2} + \frac{x}{2} \cdot 1 + 1 \frac{x}{2} + 1 \cdot 1$

Step 9.3.1.3.1.1.5

Add $1$ and $1$ .

$2 x + 4 = \frac{x^{2}}{2 \cdot 2} + \frac{x}{2} \cdot 1 + 1 \frac{x}{2} + 1 \cdot 1$

Step 9.3.1.3.1.1.6

Multiply $2$ by $2$ .

$2 x + 4 = \frac{x^{2}}{4} + \frac{x}{2} \cdot 1 + 1 \frac{x}{2} + 1 \cdot 1$

Step 9.3.1.3.1.2

Multiply $\frac{x}{2}$ by $1$ .

$2 x + 4 = \frac{x^{2}}{4} + \frac{x}{2} + 1 \frac{x}{2} + 1 \cdot 1$

Step 9.3.1.3.1.3

Multiply $\frac{x}{2}$ by $1$ .

$2 x + 4 = \frac{x^{2}}{4} + \frac{x}{2} + \frac{x}{2} + 1 \cdot 1$

Step 9.3.1.3.1.4

Multiply $1$ by $1$ .

$2 x + 4 = \frac{x^{2}}{4} + \frac{x}{2} + \frac{x}{2} + 1$

Step 9.3.1.3.2

Add $\frac{x}{2}$ and $\frac{x}{2}$ .

$2 x + 4 = \frac{x^{2}}{4} + 2 \frac{x}{2} + 1$

Step 9.3.1.4

Cancel the common factor of $2$ .

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Step 9.3.1.4.1

Cancel the common factor.

$2 x + 4 = \frac{x^{2}}{4} + 2 \frac{x}{2} + 1$

Step 9.3.1.4.2

Rewrite the expression.

$2 x + 4 = \frac{x^{2}}{4} + x + 1$

Step 10

Solve for $x$ .

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Step 10.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{x^{2}}{4} + x + 1 = 2 x + 4$

Step 10.2

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

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Step 10.2.1

Subtract $2 x$ from both sides of the equation.

$\frac{x^{2}}{4} + x + 1 - 2 x = 4$

Step 10.2.2

Subtract $2 x$ from $x$ .

$\frac{x^{2}}{4} - x + 1 = 4$

Step 10.3

Multiply each term in $\frac{x^{2}}{4} - x + 1 = 4$ by $4$ to eliminate the fractions.

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Step 10.3.1

Multiply each term in $\frac{x^{2}}{4} - x + 1 = 4$ by $4$ .

$\frac{x^{2}}{4} \cdot 4 - x \cdot 4 + 1 \cdot 4 = 4 \cdot 4$

Step 10.3.2

Simplify the left side.

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Step 10.3.2.1

Simplify each term.

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Step 10.3.2.1.1

Cancel the common factor of $4$ .

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Step 10.3.2.1.1.1

Cancel the common factor.

$\frac{x^{2}}{4} \cdot 4 - x \cdot 4 + 1 \cdot 4 = 4 \cdot 4$

Step 10.3.2.1.1.2

Rewrite the expression.

$x^{2} - x \cdot 4 + 1 \cdot 4 = 4 \cdot 4$

Step 10.3.2.1.2

Multiply $4$ by $- 1$ .

$x^{2} - 4 x + 1 \cdot 4 = 4 \cdot 4$

Step 10.3.2.1.3

Multiply $4$ by $1$ .

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

Step 10.3.3

Simplify the right side.

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Step 10.3.3.1

Multiply $4$ by $4$ .

$x^{2} - 4 x + 4 = 16$

Step 10.4

Subtract $16$ from both sides of the equation.

$x^{2} - 4 x + 4 - 16 = 0$

Step 10.5

Subtract $16$ from $4$ .

$x^{2} - 4 x - 12 = 0$

Step 10.6

Factor $x^{2} - 4 x - 12$ using the AC method.

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Step 10.6.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Step 10.6.2

Write the factored form using these integers.

$(x - 6) (x + 2) = 0$

Step 10.7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 6 = 0$

$x + 2 = 0$

Step 10.8

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 10.8.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 10.8.2

Add $6$ to both sides of the equation.

$x = 6$

Step 10.9

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 10.9.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 10.9.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 10.10

The final solution is all the values that make $(x - 6) (x + 2) = 0$ true.

$x = 6, - 2$