Algebra Examples

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

Step 1

Subtract $\sqrt{x - 7}$ from both sides of the equation.

$\sqrt{x} = 1 - \sqrt{x - 7}$

Step 2

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

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

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(1 - \sqrt{x - 7})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(1 - \sqrt{x - 7})}^{2}$ as $(1 - \sqrt{x - 7}) (1 - \sqrt{x - 7})$ .

$x = (1 - \sqrt{x - 7}) (1 - \sqrt{x - 7})$

Step 3.3.1.2

Expand $(1 - \sqrt{x - 7}) (1 - \sqrt{x - 7})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$x = 1 (1 - \sqrt{x - 7}) - \sqrt{x - 7} (1 - \sqrt{x - 7})$

Step 3.3.1.2.2

Apply the distributive property.

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

Step 3.3.1.2.3

Apply the distributive property.

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

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 $1$ .

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

Step 3.3.1.3.1.2

Multiply $- \sqrt{x - 7}$ by $1$ .

$x = 1 - \sqrt{x - 7} - \sqrt{x - 7} \cdot 1 - \sqrt{x - 7} (- \sqrt{x - 7})$

Step 3.3.1.3.1.3

Multiply $- 1$ by $1$ .

$x = 1 - \sqrt{x - 7} - \sqrt{x - 7} - \sqrt{x - 7} (- \sqrt{x - 7})$

Step 3.3.1.3.1.4

Multiply $- \sqrt{x - 7} (- \sqrt{x - 7})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 3.3.1.3.1.4.2

Multiply $\sqrt{x - 7}$ by $1$ .

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

Step 3.3.1.3.1.4.3

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

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

Step 3.3.1.3.1.4.4

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

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

Step 3.3.1.3.1.4.5

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

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

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

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

Step 3.3.1.3.1.5

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

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Step 3.3.1.3.1.5.1

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

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

Step 3.3.1.3.1.5.2

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

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

Step 3.3.1.3.1.5.3

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

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

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.

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

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

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

Step 3.3.1.3.1.5.5

Simplify.

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

Step 3.3.1.3.2

Subtract $7$ from $1$ .

$x = - 6 - \sqrt{x - 7} - \sqrt{x - 7} + x$

Step 3.3.1.3.3

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

$x = - 6 - 2 \sqrt{x - 7} + x$

Step 4

Solve for $- 2 \sqrt{x - 7}$ .

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Step 4.1

Rewrite the equation as $- 6 - 2 \sqrt{x - 7} + x = x$ .

$- 6 - 2 \sqrt{x - 7} + x = x$

Step 4.2

Move all terms not containing $- 2 \sqrt{x - 7}$ to the right side of the equation.

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Step 4.2.1

Add $6$ to both sides of the equation.

$- 2 \sqrt{x - 7} + x = x + 6$

Step 4.2.2

Subtract $x$ from both sides of the equation.

$- 2 \sqrt{x - 7} = x + 6 - x$

Step 4.2.3

Combine the opposite terms in $x + 6 - x$ .

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Step 4.2.3.1

Subtract $x$ from $x$ .

$- 2 \sqrt{x - 7} = 0 + 6$

Step 4.2.3.2

Add $0$ and $6$ .

$- 2 \sqrt{x - 7} = 6$

Step 5

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

${(- 2 \sqrt{x - 7})}^{2} = 6^{2}$

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

${(- 2 {(x - 7)}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

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

${(- 2)}^{2} {({(x - 7)}^{\frac{1}{2}})}^{2} = 6^{2}$

Step 6.2.1.2

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

$4 {({(x - 7)}^{\frac{1}{2}})}^{2} = 6^{2}$

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

$4 {(x - 7)}^{\frac{1}{2} \cdot 2} = 6^{2}$

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.

$4 {(x - 7)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$4 {(x - 7)}^{1} = 6^{2}$

Step 6.2.1.4

Simplify.

$4 (x - 7) = 6^{2}$

Step 6.2.1.5

Apply the distributive property.

$4 x + 4 \cdot - 7 = 6^{2}$

Step 6.2.1.6

Multiply $4$ by $- 7$ .

$4 x - 28 = 6^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Raise $6$ to the power of $2$ .

$4 x - 28 = 36$

Step 7

Solve for $x$ .

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Step 7.1

Move all terms not containing $x$ to the right side of the equation.

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Step 7.1.1

Add $28$ to both sides of the equation.

$4 x = 36 + 28$

Step 7.1.2

Add $36$ and $28$ .

$4 x = 64$

Step 7.2

Divide each term in $4 x = 64$ by $4$ and simplify.

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Step 7.2.1

Divide each term in $4 x = 64$ by $4$ .

$\frac{4 x}{4} = \frac{64}{4}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $4$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{64}{4}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{64}{4}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $64$ by $4$ .

$x = 16$

Step 8

Exclude the solutions that do not make $\sqrt{x} + \sqrt{x - 7} = 1$ true.

$No solution$