Solve for b square root of b-6+ square root of b=3

$\sqrt{b - 6} + \sqrt{b} = 3$

Step 1

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

$\sqrt{b - 6} = 3 - \sqrt{b}$

Step 2

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

${\sqrt{b - 6}}^{2} = {(3 - \sqrt{b})}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{b - 6}$ as ${(b - 6)}^{\frac{1}{2}}$ .

${({(b - 6)}^{\frac{1}{2}})}^{2} = {(3 - \sqrt{b})}^{2}$

Simplify the left side.

Simplify ${({(b - 6)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(b - 6)}^{\frac{1}{2}})}^{2}$ .

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

${(b - 6)}^{\frac{1}{2} \cdot 2} = {(3 - \sqrt{b})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(b - 6)}^{\frac{1}{2} \cdot 2} = {(3 - \sqrt{b})}^{2}$

Rewrite the expression.

${(b - 6)}^{1} = {(3 - \sqrt{b})}^{2}$

Simplify.

$b - 6 = {(3 - \sqrt{b})}^{2}$

Simplify the right side.

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

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

$b - 6 = (3 - \sqrt{b}) (3 - \sqrt{b})$

Expand $(3 - \sqrt{b}) (3 - \sqrt{b})$ using the FOIL Method.

Apply the distributive property.

$b - 6 = 3 (3 - \sqrt{b}) - \sqrt{b} (3 - \sqrt{b})$

Apply the distributive property.

$b - 6 = 3 \cdot 3 + 3 (- \sqrt{b}) - \sqrt{b} (3 - \sqrt{b})$

Apply the distributive property.

$b - 6 = 3 \cdot 3 + 3 (- \sqrt{b}) - \sqrt{b} \cdot 3 - \sqrt{b} (- \sqrt{b})$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$b - 6 = 9 + 3 (- \sqrt{b}) - \sqrt{b} \cdot 3 - \sqrt{b} (- \sqrt{b})$

Multiply $- 1$ by $3$ .

$b - 6 = 9 - 3 \sqrt{b} - \sqrt{b} \cdot 3 - \sqrt{b} (- \sqrt{b})$

Multiply $3$ by $- 1$ .

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} - \sqrt{b} (- \sqrt{b})$

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

Multiply $- 1$ by $- 1$ .

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + 1 \sqrt{b} \sqrt{b}$

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

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + \sqrt{b} \sqrt{b}$

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

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + {\sqrt{b}}^{1} \sqrt{b}$

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

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + {\sqrt{b}}^{1} {\sqrt{b}}^{1}$

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

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + {\sqrt{b}}^{1 + 1}$

Add $1$ and $1$ .

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + {\sqrt{b}}^{2}$

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

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

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + {(b^{\frac{1}{2}})}^{2}$

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

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + b^{\frac{1}{2} \cdot 2}$

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

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + b^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + b^{\frac{2}{2}}$

Rewrite the expression.

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + b^{1}$

Simplify.

$b - 6 = 9 - 3 \sqrt{b} - 3 \sqrt{b} + b$

Subtract $3 \sqrt{b}$ from $- 3 \sqrt{b}$ .

$b - 6 = 9 - 6 \sqrt{b} + b$

Step 4

Solve for $- 6 \sqrt{b}$ .

Rewrite the equation as $9 - 6 \sqrt{b} + b = b - 6$ .

$9 - 6 \sqrt{b} + b = b - 6$

Move all terms not containing $- 6 \sqrt{b}$ to the right side of the equation.

Subtract $9$ from both sides of the equation.

$- 6 \sqrt{b} + b = b - 6 - 9$

Subtract $b$ from both sides of the equation.

$- 6 \sqrt{b} = b - 6 - 9 - b$

Combine the opposite terms in $b - 6 - 9 - b$ .

Subtract $b$ from $b$ .

$- 6 \sqrt{b} = 0 - 6 - 9$

Subtract $6$ from $0$ .

$- 6 \sqrt{b} = - 6 - 9$

Subtract $9$ from $- 6$ .

$- 6 \sqrt{b} = - 15$

Step 5

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

${(- 6 \sqrt{b})}^{2} = {(- 15)}^{2}$

Step 6

Simplify each side of the equation.

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

${(- 6 b^{\frac{1}{2}})}^{2} = {(- 15)}^{2}$

Simplify the left side.

Simplify ${(- 6 b^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 6 b^{\frac{1}{2}}$ .

${(- 6)}^{2} {(b^{\frac{1}{2}})}^{2} = {(- 15)}^{2}$

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

$36 {(b^{\frac{1}{2}})}^{2} = {(- 15)}^{2}$

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

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

$36 b^{\frac{1}{2} \cdot 2} = {(- 15)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 b^{\frac{1}{2} \cdot 2} = {(- 15)}^{2}$

Rewrite the expression.

$36 b^{1} = {(- 15)}^{2}$

Simplify.

$36 b = {(- 15)}^{2}$

Simplify the right side.

Raise $- 15$ to the power of $2$ .

$36 b = 225$

Step 7

Divide each term in $36 b = 225$ by $36$ and simplify.

Divide each term in $36 b = 225$ by $36$ .

$\frac{36 b}{36} = \frac{225}{36}$

Simplify the left side.

Cancel the common factor of $36$ .

Cancel the common factor.

$\frac{36 b}{36} = \frac{225}{36}$

Divide $b$ by $1$ .

$b = \frac{225}{36}$

Simplify the right side.

Cancel the common factor of $225$ and $36$ .

Factor $9$ out of $225$ .

$b = \frac{9 (25)}{36}$

Cancel the common factors.

Factor $9$ out of $36$ .

$b = \frac{9 \cdot 25}{9 \cdot 4}$

Cancel the common factor.

$b = \frac{9 \cdot 25}{9 \cdot 4}$

Rewrite the expression.

$b = \frac{25}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$b = \frac{25}{4}$

Decimal Form:

$b = 6.25$

Mixed Number Form:

$b = 6 \frac{1}{4}$

Solve for b square root of b-6+ square root of b=3

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