Solve by Factoring square root of b-17+ square root of b=1

$\sqrt{b - 17} + \sqrt{b} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\sqrt{b - 17} + \sqrt{b} - 1 = 0$

Step 2

Move all terms not containing $\sqrt{b - 17}$ to the right side of the equation.

Subtract $\sqrt{b}$ from both sides of the equation.

$\sqrt{b - 17} - 1 = - \sqrt{b}$

Add $1$ to both sides of the equation.

$\sqrt{b - 17} = - \sqrt{b} + 1$

Step 3

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

${\sqrt{b - 17}}^{2} = {(- \sqrt{b} + 1)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{b - 17}$ as ${(b - 17)}^{\frac{1}{2}}$ .

${({(b - 17)}^{\frac{1}{2}})}^{2} = {(- \sqrt{b} + 1)}^{2}$

Simplify the left side.

Simplify ${({(b - 17)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(b - 17)}^{\frac{1}{2}})}^{2}$ .

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

${(b - 17)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{b} + 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(b - 17)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{b} + 1)}^{2}$

Rewrite the expression.

${(b - 17)}^{1} = {(- \sqrt{b} + 1)}^{2}$

Simplify.

$b - 17 = {(- \sqrt{b} + 1)}^{2}$

Simplify the right side.

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

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

$b - 17 = (- \sqrt{b} + 1) (- \sqrt{b} + 1)$

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

Apply the distributive property.

$b - 17 = - \sqrt{b} (- \sqrt{b} + 1) + 1 (- \sqrt{b} + 1)$

Apply the distributive property.

$b - 17 = - \sqrt{b} (- \sqrt{b}) - \sqrt{b} \cdot 1 + 1 (- \sqrt{b} + 1)$

Apply the distributive property.

$b - 17 = - \sqrt{b} (- \sqrt{b}) - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Multiply $- \sqrt{b} (- \sqrt{b})$ .

Multiply $- 1$ by $- 1$ .

$b - 17 = 1 \sqrt{b} \sqrt{b} - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

Multiply $\sqrt{b}$ by $1$ .

$b - 17 = \sqrt{b} \sqrt{b} - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

Raise $\sqrt{b}$ to the power of $1$ .

$b - 17 = {\sqrt{b}}^{1} \sqrt{b} - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

Raise $\sqrt{b}$ to the power of $1$ .

$b - 17 = {\sqrt{b}}^{1} {\sqrt{b}}^{1} - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

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

$b - 17 = {\sqrt{b}}^{1 + 1} - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

Add $1$ and $1$ .

$b - 17 = {\sqrt{b}}^{2} - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

Rewrite ${\sqrt{b}}^{2}$ as $b$ .

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

$b - 17 = {(b^{\frac{1}{2}})}^{2} - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

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

$b - 17 = b^{\frac{1}{2} \cdot 2} - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

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

$b - 17 = b^{\frac{2}{2}} - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$b - 17 = b^{\frac{2}{2}} - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

Rewrite the expression.

$b - 17 = b^{1} - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

Simplify.

$b - 17 = b - \sqrt{b} \cdot 1 + 1 (- \sqrt{b}) + 1 \cdot 1$

Multiply $- 1$ by $1$ .

$b - 17 = b - \sqrt{b} + 1 (- \sqrt{b}) + 1 \cdot 1$

Multiply $- \sqrt{b}$ by $1$ .

$b - 17 = b - \sqrt{b} - \sqrt{b} + 1 \cdot 1$

Multiply $1$ by $1$ .

$b - 17 = b - \sqrt{b} - \sqrt{b} + 1$

Subtract $\sqrt{b}$ from $- \sqrt{b}$ .

$b - 17 = b - 2 \sqrt{b} + 1$

Step 5

Solve for $- 2 \sqrt{b}$ .

Rewrite the equation as $b - 2 \sqrt{b} + 1 = b - 17$ .

$b - 2 \sqrt{b} + 1 = b - 17$

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

Subtract $b$ from both sides of the equation.

$- 2 \sqrt{b} + 1 = b - 17 - b$

Subtract $1$ from both sides of the equation.

$- 2 \sqrt{b} = b - 17 - b - 1$

Combine the opposite terms in $b - 17 - b - 1$ .

Subtract $b$ from $b$ .

$- 2 \sqrt{b} = 0 - 17 - 1$

Subtract $17$ from $0$ .

$- 2 \sqrt{b} = - 17 - 1$

Subtract $1$ from $- 17$ .

$- 2 \sqrt{b} = - 18$

Step 6

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

${(- 2 \sqrt{b})}^{2} = {(- 18)}^{2}$

Step 7

Simplify each side of the equation.

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

${(- 2 b^{\frac{1}{2}})}^{2} = {(- 18)}^{2}$

Simplify the left side.

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

Apply the product rule to $- 2 b^{\frac{1}{2}}$ .

${(- 2)}^{2} {(b^{\frac{1}{2}})}^{2} = {(- 18)}^{2}$

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

$4 {(b^{\frac{1}{2}})}^{2} = {(- 18)}^{2}$

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

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

$4 b^{\frac{1}{2} \cdot 2} = {(- 18)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 b^{\frac{1}{2} \cdot 2} = {(- 18)}^{2}$

Rewrite the expression.

$4 b^{1} = {(- 18)}^{2}$

Simplify.

$4 b = {(- 18)}^{2}$

Simplify the right side.

Raise $- 18$ to the power of $2$ .

$4 b = 324$

Step 8

Divide each term in $4 b = 324$ by $4$ and simplify.

Divide each term in $4 b = 324$ by $4$ .

$\frac{4 b}{4} = \frac{324}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 b}{4} = \frac{324}{4}$

Divide $b$ by $1$ .

$b = \frac{324}{4}$

Simplify the right side.

Divide $324$ by $4$ .

$b = 81$

Step 9

Exclude the solutions that do not make $\sqrt{b - 17} + \sqrt{b} - 1 = 0$ true.

$No solution$

Solve by Factoring square root of b-17+ square root of b=1

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