Precalculus Examples

$\sqrt{2 \sqrt{2 x + 3}} = \sqrt{x + 3}$

Step 1

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

${\sqrt{2 \sqrt{2 x + 3}}}^{2} = {\sqrt{x + 3}}^{2}$

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 $\sqrt{2 \sqrt{2 x + 3}}$ as ${(2 \sqrt{2 x + 3})}^{\frac{1}{2}}$ .

${({(2 \sqrt{2 x + 3})}^{\frac{1}{2}})}^{2} = {\sqrt{x + 3}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(2 \sqrt{2 x + 3})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(2 \sqrt{2 x + 3})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(2 \sqrt{2 x + 3})}^{\frac{1}{2} \cdot 2} = {\sqrt{x + 3}}^{2}$

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.

${(2 \sqrt{2 x + 3})}^{\frac{1}{2} \cdot 2} = {\sqrt{x + 3}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 \sqrt{2 x + 3})}^{1} = {\sqrt{x + 3}}^{2}$

Step 2.2.1.2

Simplify.

$2 \sqrt{2 x + 3} = {\sqrt{x + 3}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt{x + 3}}^{2}$ as $x + 3$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

$2 \sqrt{2 x + 3} = {({(x + 3)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2

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

$2 \sqrt{2 x + 3} = {(x + 3)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3

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

$2 \sqrt{2 x + 3} = {(x + 3)}^{\frac{2}{2}}$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

$2 \sqrt{2 x + 3} = {(x + 3)}^{\frac{2}{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$2 \sqrt{2 x + 3} = {(x + 3)}^{1}$

Step 2.3.1.5

Simplify.

$2 \sqrt{2 x + 3} = x + 3$

Step 3

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

${(2 \sqrt{2 x + 3})}^{2} = {(x + 3)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 3}$ as ${(2 x + 3)}^{\frac{1}{2}}$ .

${(2 {(2 x + 3)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(2 {(2 x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Apply the product rule to $2 {(2 x + 3)}^{\frac{1}{2}}$ .

$2^{2} {({(2 x + 3)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Step 4.2.1.2

Raise $2$ to the power of $2$ .

$4 {({(2 x + 3)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Step 4.2.1.3

Multiply the exponents in ${({(2 x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.3.1

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

$4 {(2 x + 3)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 4.2.1.3.2

Cancel the common factor of $2$ .

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Step 4.2.1.3.2.1

Cancel the common factor.

$4 {(2 x + 3)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 4.2.1.3.2.2

Rewrite the expression.

$4 {(2 x + 3)}^{1} = {(x + 3)}^{2}$

Step 4.2.1.4

Simplify.

$4 (2 x + 3) = {(x + 3)}^{2}$

Step 4.2.1.5

Apply the distributive property.

$4 (2 x) + 4 \cdot 3 = {(x + 3)}^{2}$

Step 4.2.1.6

Multiply.

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Step 4.2.1.6.1

Multiply $2$ by $4$ .

$8 x + 4 \cdot 3 = {(x + 3)}^{2}$

Step 4.2.1.6.2

Multiply $4$ by $3$ .

$8 x + 12 = {(x + 3)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(x + 3)}^{2}$ .

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Step 4.3.1.1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$8 x + 12 = (x + 3) (x + 3)$

Step 4.3.1.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$8 x + 12 = x (x + 3) + 3 (x + 3)$

Step 4.3.1.2.2

Apply the distributive property.

$8 x + 12 = x \cdot x + x \cdot 3 + 3 (x + 3)$

Step 4.3.1.2.3

Apply the distributive property.

$8 x + 12 = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Multiply $x$ by $x$ .

$8 x + 12 = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Step 4.3.1.3.1.2

Move $3$ to the left of $x$ .

$8 x + 12 = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Step 4.3.1.3.1.3

Multiply $3$ by $3$ .

$8 x + 12 = x^{2} + 3 x + 3 x + 9$

Step 4.3.1.3.2

Add $3 x$ and $3 x$ .

$8 x + 12 = x^{2} + 6 x + 9$

Step 5

Solve for $x$ .

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Step 5.1

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

$x^{2} + 6 x + 9 = 8 x + 12$

Step 5.2

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

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Step 5.2.1

Subtract $8 x$ from both sides of the equation.

$x^{2} + 6 x + 9 - 8 x = 12$

Step 5.2.2

Subtract $8 x$ from $6 x$ .

$x^{2} - 2 x + 9 = 12$

Step 5.3

Subtract $12$ from both sides of the equation.

$x^{2} - 2 x + 9 - 12 = 0$

Step 5.4

Subtract $12$ from $9$ .

$x^{2} - 2 x - 3 = 0$

Step 5.5

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 5.5.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 5.5.2

Write the factored form using these integers.

$(x - 3) (x + 1) = 0$

Step 5.6

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

$x - 3 = 0$

$x + 1 = 0$

Step 5.7

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 5.7.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 5.7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5.8

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 5.8.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 5.8.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5.9

The final solution is all the values that make $(x - 3) (x + 1) = 0$ true.

$x = 3, - 1$