Algebra Examples

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

Step 1

Subtract $2$ from both sides of the equation.

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

Step 2

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

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

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

Expand $(\sqrt{x + 7} - 2) (\sqrt{x + 7} - 2)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

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

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

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

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 $\sqrt{x + 7} \sqrt{x + 7}$ .

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Step 3.3.1.3.1.1.1

Raise $\sqrt{x + 7}$ to the power of $1$ .

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

Step 3.3.1.3.1.1.2

Raise $\sqrt{x + 7}$ to the power of $1$ .

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

Step 3.3.1.3.1.1.3

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

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

Step 3.3.1.3.1.1.4

Add $1$ and $1$ .

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

Step 3.3.1.3.1.2

Rewrite ${\sqrt{x + 7}}^{2}$ as $x + 7$ .

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Step 3.3.1.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 7}$ as ${(x + 7)}^{\frac{1}{2}}$ .

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

Step 3.3.1.3.1.2.2

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

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

Step 3.3.1.3.1.2.3

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

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

Step 3.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.2.4.1

Cancel the common factor.

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

Step 3.3.1.3.1.2.4.2

Rewrite the expression.

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

Step 3.3.1.3.1.2.5

Simplify.

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

Step 3.3.1.3.1.3

Move $- 2$ to the left of $\sqrt{x + 7}$ .

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

Step 3.3.1.3.1.4

Multiply $- 2$ by $- 2$ .

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

Step 3.3.1.3.2

Add $7$ and $4$ .

$2 x - 3 = x + 11 - 2 \sqrt{x + 7} - 2 \sqrt{x + 7}$

Step 3.3.1.3.3

Subtract $2 \sqrt{x + 7}$ from $- 2 \sqrt{x + 7}$ .

$2 x - 3 = x + 11 - 4 \sqrt{x + 7}$

Step 4

Solve for $- 4 \sqrt{x + 7}$ .

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Step 4.1

Rewrite the equation as $x + 11 - 4 \sqrt{x + 7} = 2 x - 3$ .

$x + 11 - 4 \sqrt{x + 7} = 2 x - 3$

Step 4.2

Move all terms not containing $- 4 \sqrt{x + 7}$ to the right side of the equation.

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Step 4.2.1

Subtract $x$ from both sides of the equation.

$11 - 4 \sqrt{x + 7} = 2 x - 3 - x$

Step 4.2.2

Subtract $11$ from both sides of the equation.

$- 4 \sqrt{x + 7} = 2 x - 3 - x - 11$

Step 4.2.3

Subtract $x$ from $2 x$ .

$- 4 \sqrt{x + 7} = x - 3 - 11$

Step 4.2.4

Subtract $11$ from $- 3$ .

$- 4 \sqrt{x + 7} = x - 14$

Step 5

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

${(- 4 \sqrt{x + 7})}^{2} = {(x - 14)}^{2}$

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

${(- 4 {(x + 7)}^{\frac{1}{2}})}^{2} = {(x - 14)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 4 {(x + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $- 4 {(x + 7)}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(x + 7)}^{\frac{1}{2}})}^{2} = {(x - 14)}^{2}$

Step 6.2.1.2

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

$16 {({(x + 7)}^{\frac{1}{2}})}^{2} = {(x - 14)}^{2}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$16 {(x + 7)}^{\frac{1}{2} \cdot 2} = {(x - 14)}^{2}$

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.

$16 {(x + 7)}^{\frac{1}{2} \cdot 2} = {(x - 14)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$16 {(x + 7)}^{1} = {(x - 14)}^{2}$

Step 6.2.1.4

Simplify.

$16 (x + 7) = {(x - 14)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$16 x + 16 \cdot 7 = {(x - 14)}^{2}$

Step 6.2.1.6

Multiply $16$ by $7$ .

$16 x + 112 = {(x - 14)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(x - 14)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(x - 14)}^{2}$ as $(x - 14) (x - 14)$ .

$16 x + 112 = (x - 14) (x - 14)$

Step 6.3.1.2

Expand $(x - 14) (x - 14)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$16 x + 112 = x (x - 14) - 14 (x - 14)$

Step 6.3.1.2.2

Apply the distributive property.

$16 x + 112 = x \cdot x + x \cdot - 14 - 14 (x - 14)$

Step 6.3.1.2.3

Apply the distributive property.

$16 x + 112 = x \cdot x + x \cdot - 14 - 14 x - 14 \cdot - 14$

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

Multiply $x$ by $x$ .

$16 x + 112 = x^{2} + x \cdot - 14 - 14 x - 14 \cdot - 14$

Step 6.3.1.3.1.2

Move $- 14$ to the left of $x$ .

$16 x + 112 = x^{2} - 14 \cdot x - 14 x - 14 \cdot - 14$

Step 6.3.1.3.1.3

Multiply $- 14$ by $- 14$ .

$16 x + 112 = x^{2} - 14 x - 14 x + 196$

Step 6.3.1.3.2

Subtract $14 x$ from $- 14 x$ .

$16 x + 112 = x^{2} - 28 x + 196$

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.

$x^{2} - 28 x + 196 = 16 x + 112$

Step 7.2

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

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Step 7.2.1

Subtract $16 x$ from both sides of the equation.

$x^{2} - 28 x + 196 - 16 x = 112$

Step 7.2.2

Subtract $16 x$ from $- 28 x$ .

$x^{2} - 44 x + 196 = 112$

Step 7.3

Subtract $112$ from both sides of the equation.

$x^{2} - 44 x + 196 - 112 = 0$

Step 7.4

Subtract $112$ from $196$ .

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

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

$x = 2$

Step 9