Solve by Factoring square root of x+ square root of x-7=1

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

Step 1

Subtract $1$ from both sides of the equation.

$\sqrt{x} + \sqrt{x - 7} - 1 = 0$

Step 2

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

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

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

Add $1$ to both sides of the equation.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Multiply $- 1$ by $- 1$ .

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Multiply $- 1$ by $1$ .

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

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

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

Multiply $1$ by $1$ .

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

Add $- 7$ and $1$ .

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

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

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

Step 5

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

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

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

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

Subtract $x$ from both sides of the equation.

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

Add $6$ to both sides of the equation.

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

Subtract $x$ from $x$ .

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

Add $0$ and $6$ .

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

Step 6

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 7

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $4$ by $- 7$ .

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

Simplify the right side.

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

$4 x - 28 = 36$

Step 8

Solve for $x$ .

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

Add $28$ to both sides of the equation.

$4 x = 36 + 28$

Add $36$ and $28$ .

$4 x = 64$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $64$ by $4$ .

$x = 16$

Step 9

Exclude the solutions that do not make $\sqrt{x} + \sqrt{x - 7} - 1 = 0$ true.

$No solution$

Solve by Factoring square root of x+ square root of x-7=1

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