Precalculus Examples

$2 \sqrt{x + 1} - \sqrt{2 x + 3} = 1$

Step 1

Add $\sqrt{2 x + 3}$ to both sides of the equation.

$2 \sqrt{x + 1} = 1 + \sqrt{2 x + 3}$

Step 2

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

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

Step 3

Simplify each side of the equation.

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

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

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

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

Step 3.2.1.3.2.2

Rewrite the expression.

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

Step 3.2.1.4

Simplify.

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

Step 3.2.1.5

Apply the distributive property.

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

Step 3.2.1.6

Multiply $4$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

Expand $(1 + \sqrt{2 x + 3}) (1 + \sqrt{2 x + 3})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

$4 x + 4 = 1 \cdot 1 + 1 \sqrt{2 x + 3} + \sqrt{2 x + 3} \cdot 1 + \sqrt{2 x + 3} \sqrt{2 x + 3}$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $1$ by $1$ .

$4 x + 4 = 1 + 1 \sqrt{2 x + 3} + \sqrt{2 x + 3} \cdot 1 + \sqrt{2 x + 3} \sqrt{2 x + 3}$

Step 3.3.1.3.1.2

Multiply $\sqrt{2 x + 3}$ by $1$ .

$4 x + 4 = 1 + \sqrt{2 x + 3} + \sqrt{2 x + 3} \cdot 1 + \sqrt{2 x + 3} \sqrt{2 x + 3}$

Step 3.3.1.3.1.3

Multiply $\sqrt{2 x + 3}$ by $1$ .

$4 x + 4 = 1 + \sqrt{2 x + 3} + \sqrt{2 x + 3} + \sqrt{2 x + 3} \sqrt{2 x + 3}$

Step 3.3.1.3.1.4

Multiply $\sqrt{2 x + 3} \sqrt{2 x + 3}$ .

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Step 3.3.1.3.1.4.1

Raise $\sqrt{2 x + 3}$ to the power of $1$ .

$4 x + 4 = 1 + \sqrt{2 x + 3} + \sqrt{2 x + 3} + {\sqrt{2 x + 3}}^{1} \sqrt{2 x + 3}$

Step 3.3.1.3.1.4.2

Raise $\sqrt{2 x + 3}$ to the power of $1$ .

$4 x + 4 = 1 + \sqrt{2 x + 3} + \sqrt{2 x + 3} + {\sqrt{2 x + 3}}^{1} {\sqrt{2 x + 3}}^{1}$

Step 3.3.1.3.1.4.3

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

$4 x + 4 = 1 + \sqrt{2 x + 3} + \sqrt{2 x + 3} + {\sqrt{2 x + 3}}^{1 + 1}$

Step 3.3.1.3.1.4.4

Add $1$ and $1$ .

$4 x + 4 = 1 + \sqrt{2 x + 3} + \sqrt{2 x + 3} + {\sqrt{2 x + 3}}^{2}$

Step 3.3.1.3.1.5

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

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Step 3.3.1.3.1.5.1

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

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

Step 3.3.1.3.1.5.2

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

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

Step 3.3.1.3.1.5.3

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

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

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

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

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

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

Step 3.3.1.3.1.5.5

Simplify.

$4 x + 4 = 1 + \sqrt{2 x + 3} + \sqrt{2 x + 3} + 2 x + 3$

Step 3.3.1.3.2

Add $1$ and $3$ .

$4 x + 4 = 4 + \sqrt{2 x + 3} + \sqrt{2 x + 3} + 2 x$

Step 3.3.1.3.3

Add $\sqrt{2 x + 3}$ and $\sqrt{2 x + 3}$ .

$4 x + 4 = 4 + 2 \sqrt{2 x + 3} + 2 x$

Step 4

Solve for $2 \sqrt{2 x + 3}$ .

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Step 4.1

Rewrite the equation as $4 + 2 \sqrt{2 x + 3} + 2 x = 4 x + 4$ .

$4 + 2 \sqrt{2 x + 3} + 2 x = 4 x + 4$

Step 4.2

Move all terms not containing $2 \sqrt{2 x + 3}$ to the right side of the equation.

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$2 \sqrt{2 x + 3} + 2 x = 4 x + 4 - 4$

Step 4.2.2

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

$2 \sqrt{2 x + 3} = 4 x + 4 - 4 - 2 x$

Step 4.2.3

Combine the opposite terms in $4 x + 4 - 4 - 2 x$ .

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Step 4.2.3.1

Subtract $4$ from $4$ .

$2 \sqrt{2 x + 3} = 4 x + 0 - 2 x$

Step 4.2.3.2

Add $4 x$ and $0$ .

$2 \sqrt{2 x + 3} = 4 x - 2 x$

Step 4.2.4

Subtract $2 x$ from $4 x$ .

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

Step 5

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

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

Step 6

Simplify each side of the equation.

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

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

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

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

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

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

Step 6.2.1.3.2.2

Rewrite the expression.

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

Step 6.2.1.4

Simplify.

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

Step 6.2.1.5

Apply the distributive property.

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

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $2$ by $4$ .

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

Step 6.2.1.6.2

Multiply $4$ by $3$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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Step 6.3.1.1

Apply the product rule to $2 x$ .

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

Step 6.3.1.2

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

$8 x + 12 = 4 x^{2}$

Step 7

Solve for $x$ .

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Step 7.1

Subtract $4 x^{2}$ from both sides of the equation.

$8 x + 12 - 4 x^{2} = 0$

Step 7.2

Factor the left side of the equation.

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Step 7.2.1

Factor $- 4$ out of $8 x + 12 - 4 x^{2}$ .

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Step 7.2.1.1

Reorder the expression.

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Step 7.2.1.1.1

Move $12$ .

$8 x - 4 x^{2} + 12 = 0$

Step 7.2.1.1.2

Reorder $8 x$ and $- 4 x^{2}$ .

$- 4 x^{2} + 8 x + 12 = 0$

Step 7.2.1.2

Factor $- 4$ out of $- 4 x^{2}$ .

$- 4 x^{2} + 8 x + 12 = 0$

Step 7.2.1.3

Factor $- 4$ out of $8 x$ .

$- 4 x^{2} - 4 (- 2 x) + 12 = 0$

Step 7.2.1.4

Factor $- 4$ out of $12$ .

$- 4 x^{2} - 4 (- 2 x) - 4 \cdot - 3 = 0$

Step 7.2.1.5

Factor $- 4$ out of $- 4 (x^{2}) - 4 (- 2 x)$ .

$- 4 (x^{2} - 2 x) - 4 \cdot - 3 = 0$

Step 7.2.1.6

Factor $- 4$ out of $- 4 (x^{2} - 2 x) - 4 (- 3)$ .

$- 4 (x^{2} - 2 x - 3) = 0$

Step 7.2.2

Factor.

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Step 7.2.2.1

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

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Step 7.2.2.1.1

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 7.2.2.1.2

Write the factored form using these integers.

$- 4 ((x - 3) (x + 1)) = 0$

Step 7.2.2.2

Remove unnecessary parentheses.

$- 4 (x - 3) (x + 1) = 0$

Step 7.3

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 7.4

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

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Step 7.4.1

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

$x - 3 = 0$

Step 7.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 7.5

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

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Step 7.5.1

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

$x + 1 = 0$

Step 7.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7.6

The final solution is all the values that make $- 4 (x - 3) (x + 1) = 0$ true.

$x = 3, - 1$

Step 8

Exclude the solutions that do not make $2 \sqrt{x + 1} - \sqrt{2 x + 3} = 1$ true.

$x = 3$