Precalculus Examples

$\sqrt{x + 7} - \sqrt{2 x - 3} = 2$

Step 1

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

$\sqrt{x + 7} = 2 + \sqrt{2 x - 3}$

Step 2

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

${\sqrt{x + 7}}^{2} = {(2 + \sqrt{2 x - 3})}^{2}$

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 $\sqrt{x + 7}$ as ${(x + 7)}^{\frac{1}{2}}$ .

${({(x + 7)}^{\frac{1}{2}})}^{2} = {(2 + \sqrt{2 x - 3})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{2 x - 3})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(x + 7)}^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{2 x - 3})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x + 7)}^{1} = {(2 + \sqrt{2 x - 3})}^{2}$

Step 3.2.1.2

Simplify.

$x + 7 = {(2 + \sqrt{2 x - 3})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

$x + 7 = (2 + \sqrt{2 x - 3}) (2 + \sqrt{2 x - 3})$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Apply the distributive property.

$x + 7 = 2 (2 + \sqrt{2 x - 3}) + \sqrt{2 x - 3} (2 + \sqrt{2 x - 3})$

Step 3.3.1.2.2

Apply the distributive property.

$x + 7 = 2 \cdot 2 + 2 \sqrt{2 x - 3} + \sqrt{2 x - 3} (2 + \sqrt{2 x - 3})$

Step 3.3.1.2.3

Apply the distributive property.

$x + 7 = 2 \cdot 2 + 2 \sqrt{2 x - 3} + \sqrt{2 x - 3} \cdot 2 + \sqrt{2 x - 3} \sqrt{2 x - 3}$

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 $2$ by $2$ .

$x + 7 = 4 + 2 \sqrt{2 x - 3} + \sqrt{2 x - 3} \cdot 2 + \sqrt{2 x - 3} \sqrt{2 x - 3}$

Step 3.3.1.3.1.2

Move $2$ to the left of $\sqrt{2 x - 3}$ .

$x + 7 = 4 + 2 \sqrt{2 x - 3} + 2 \cdot \sqrt{2 x - 3} + \sqrt{2 x - 3} \sqrt{2 x - 3}$

Step 3.3.1.3.1.3

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

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Step 3.3.1.3.1.3.1

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

$x + 7 = 4 + 2 \sqrt{2 x - 3} + 2 \sqrt{2 x - 3} + {\sqrt{2 x - 3}}^{1} \sqrt{2 x - 3}$

Step 3.3.1.3.1.3.2

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

$x + 7 = 4 + 2 \sqrt{2 x - 3} + 2 \sqrt{2 x - 3} + {\sqrt{2 x - 3}}^{1} {\sqrt{2 x - 3}}^{1}$

Step 3.3.1.3.1.3.3

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

$x + 7 = 4 + 2 \sqrt{2 x - 3} + 2 \sqrt{2 x - 3} + {\sqrt{2 x - 3}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$x + 7 = 4 + 2 \sqrt{2 x - 3} + 2 \sqrt{2 x - 3} + {\sqrt{2 x - 3}}^{2}$

Step 3.3.1.3.1.4

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

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Step 3.3.1.3.1.4.1

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

$x + 7 = 4 + 2 \sqrt{2 x - 3} + 2 \sqrt{2 x - 3} + {({(2 x - 3)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

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

$x + 7 = 4 + 2 \sqrt{2 x - 3} + 2 \sqrt{2 x - 3} + {(2 x - 3)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

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

$x + 7 = 4 + 2 \sqrt{2 x - 3} + 2 \sqrt{2 x - 3} + {(2 x - 3)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$x + 7 = 4 + 2 \sqrt{2 x - 3} + 2 \sqrt{2 x - 3} + {(2 x - 3)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$x + 7 = 4 + 2 \sqrt{2 x - 3} + 2 \sqrt{2 x - 3} + {(2 x - 3)}^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$x + 7 = 4 + 2 \sqrt{2 x - 3} + 2 \sqrt{2 x - 3} + 2 x - 3$

Step 3.3.1.3.2

Subtract $3$ from $4$ .

$x + 7 = 1 + 2 \sqrt{2 x - 3} + 2 \sqrt{2 x - 3} + 2 x$

Step 3.3.1.3.3

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

$x + 7 = 1 + 4 \sqrt{2 x - 3} + 2 x$

Step 4

Solve for $4 \sqrt{2 x - 3}$ .

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Step 4.1

Rewrite the equation as $1 + 4 \sqrt{2 x - 3} + 2 x = x + 7$ .

$1 + 4 \sqrt{2 x - 3} + 2 x = x + 7$

Step 4.2

Move all terms not containing $4 \sqrt{2 x - 3}$ to the right side of the equation.

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$4 \sqrt{2 x - 3} + 2 x = x + 7 - 1$

Step 4.2.2

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

$4 \sqrt{2 x - 3} = x + 7 - 1 - 2 x$

Step 4.2.3

Subtract $2 x$ from $x$ .

$4 \sqrt{2 x - 3} = - x + 7 - 1$

Step 4.2.4

Subtract $1$ from $7$ .

$4 \sqrt{2 x - 3} = - x + 6$

Step 5

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

${(4 \sqrt{2 x - 3})}^{2} = {(- x + 6)}^{2}$

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

${(4 {(2 x - 3)}^{\frac{1}{2}})}^{2} = {(- x + 6)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(4 {(2 x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $4 {(2 x - 3)}^{\frac{1}{2}}$ .

$4^{2} {({(2 x - 3)}^{\frac{1}{2}})}^{2} = {(- x + 6)}^{2}$

Step 6.2.1.2

Raise $4$ to the power of $2$ .

$16 {({(2 x - 3)}^{\frac{1}{2}})}^{2} = {(- x + 6)}^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$16 {(2 x - 3)}^{\frac{1}{2} \cdot 2} = {(- x + 6)}^{2}$

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.

$16 {(2 x - 3)}^{\frac{1}{2} \cdot 2} = {(- x + 6)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$16 {(2 x - 3)}^{1} = {(- x + 6)}^{2}$

Step 6.2.1.4

Simplify.

$16 (2 x - 3) = {(- x + 6)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$16 (2 x) + 16 \cdot - 3 = {(- x + 6)}^{2}$

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $2$ by $16$ .

$32 x + 16 \cdot - 3 = {(- x + 6)}^{2}$

Step 6.2.1.6.2

Multiply $16$ by $- 3$ .

$32 x - 48 = {(- x + 6)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(- x + 6)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(- x + 6)}^{2}$ as $(- x + 6) (- x + 6)$ .

$32 x - 48 = (- x + 6) (- x + 6)$

Step 6.3.1.2

Expand $(- x + 6) (- x + 6)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$32 x - 48 = - x (- x + 6) + 6 (- x + 6)$

Step 6.3.1.2.2

Apply the distributive property.

$32 x - 48 = - x (- x) - x \cdot 6 + 6 (- x + 6)$

Step 6.3.1.2.3

Apply the distributive property.

$32 x - 48 = - x (- x) - x \cdot 6 + 6 (- x) + 6 \cdot 6$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$32 x - 48 = - 1 \cdot - 1 x \cdot x - x \cdot 6 + 6 (- x) + 6 \cdot 6$

Step 6.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1.3.1.2.1

Move $x$ .

$32 x - 48 = - 1 \cdot - 1 (x \cdot x) - x \cdot 6 + 6 (- x) + 6 \cdot 6$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$32 x - 48 = - 1 \cdot - 1 x^{2} - x \cdot 6 + 6 (- x) + 6 \cdot 6$

Step 6.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$32 x - 48 = 1 x^{2} - x \cdot 6 + 6 (- x) + 6 \cdot 6$

Step 6.3.1.3.1.4

Multiply $x^{2}$ by $1$ .

$32 x - 48 = x^{2} - x \cdot 6 + 6 (- x) + 6 \cdot 6$

Step 6.3.1.3.1.5

Multiply $6$ by $- 1$ .

$32 x - 48 = x^{2} - 6 x + 6 (- x) + 6 \cdot 6$

Step 6.3.1.3.1.6

Multiply $- 1$ by $6$ .

$32 x - 48 = x^{2} - 6 x - 6 x + 6 \cdot 6$

Step 6.3.1.3.1.7

Multiply $6$ by $6$ .

$32 x - 48 = x^{2} - 6 x - 6 x + 36$

Step 6.3.1.3.2

Subtract $6 x$ from $- 6 x$ .

$32 x - 48 = x^{2} - 12 x + 36$

Step 7

Solve for $x$ .

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Step 7.1

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

$x^{2} - 12 x + 36 = 32 x - 48$

Step 7.2

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

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Step 7.2.1

Subtract $32 x$ from both sides of the equation.

$x^{2} - 12 x + 36 - 32 x = - 48$

Step 7.2.2

Subtract $32 x$ from $- 12 x$ .

$x^{2} - 44 x + 36 = - 48$

Step 7.3

Add $48$ to both sides of the equation.

$x^{2} - 44 x + 36 + 48 = 0$

Step 7.4

Add $36$ and $48$ .

$x^{2} - 44 x + 84 = 0$

Step 7.5

Factor $x^{2} - 44 x + 84$ using the AC method.

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Step 7.5.1

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 $84$ and whose sum is $- 44$ .

$- 42, - 2$

Step 7.5.2

Write the factored form using these integers.

$(x - 42) (x - 2) = 0$

Step 7.6

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

$x - 42 = 0$

$x - 2 = 0$

Step 7.7

Set $x - 42$ equal to $0$ and solve for $x$ .

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Step 7.7.1

Set $x - 42$ equal to $0$ .

$x - 42 = 0$

Step 7.7.2

Add $42$ to both sides of the equation.

$x = 42$

Step 7.8

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

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Step 7.8.1

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

$x - 2 = 0$

Step 7.8.2

Add $2$ to both sides of the equation.

$x = 2$

Step 7.9

The final solution is all the values that make $(x - 42) (x - 2) = 0$ true.

$x = 42, 2$

Step 8

Exclude the solutions that do not make $\sqrt{x + 7} - \sqrt{2 x - 3} = 2$ true.

$x = 2$