Solve for b square root of 6-b=2b square root of 6-b

$\sqrt{6 - b} = 2 b \sqrt{6 - b}$

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

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

Simplify each side of the equation.

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $2 b \sqrt{6 - b}$ .

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

Apply the product rule to $2 b$ .

$6 - b = 2^{2} b^{2} {\sqrt{6 - b}}^{2}$

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

$6 - b = 4 b^{2} {\sqrt{6 - b}}^{2}$

Rewrite ${\sqrt{6 - b}}^{2}$ as $6 - b$ .

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

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

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

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

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

$6 - b = 4 b^{2} {(6 - b)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$6 - b = 4 b^{2} {(6 - b)}^{\frac{2}{2}}$

Divide $1$ by $1$ .

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

Simplify.

$6 - b = 4 b^{2} (6 - b)$

Apply the distributive property.

$6 - b = 4 b^{2} \cdot 6 + 4 b^{2} (- b)$

Multiply $6$ by $4$ .

$6 - b = 24 b^{2} + 4 b^{2} (- b)$

Rewrite using the commutative property of multiplication.

$6 - b = 24 b^{2} + 4 \cdot - 1 (b^{2} b)$

Simplify each term.

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Multiply $b^{2}$ by $b$ by adding the exponents.

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Multiply $b^{2}$ by $b$ .

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Raise $b$ to the power of $1$ .

$6 - b = 24 b^{2} + 4 \cdot - 1 (b^{2} b^{1})$

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

$6 - b = 24 b^{2} + 4 \cdot - 1 b^{2 + 1}$

Add $2$ and $1$ .

$6 - b = 24 b^{2} + 4 \cdot - 1 b^{3}$

Multiply $4$ by $- 1$ .

$6 - b = 24 b^{2} - 4 b^{3}$

Solve for $b$ .

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Since $b$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$24 b^{2} - 4 b^{3} = 6 - b$

Add $b$ to both sides of the equation.

$24 b^{2} - 4 b^{3} + b = 6$

Move $6$ to the left side of the equation by subtracting it from both sides.

$24 b^{2} - 4 b^{3} + b - 6 = 0$

Factor the left side of the equation.

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Reorder terms.

$- 4 b^{3} + 24 b^{2} + b - 6 = 0$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$(- 4 b^{3} + 24 b^{2}) + b - 6 = 0$

Factor out the greatest common factor (GCF) from each group.

$4 b^{2} (- b + 6) - 1 (- b + 6) = 0$

Factor the polynomial by factoring out the greatest common factor, $- b + 6$ .

$(- b + 6) (4 b^{2} - 1) = 0$

Rewrite $4 b^{2}$ as ${(2 b)}^{2}$ .

$(- b + 6) ({(2 b)}^{2} - 1) = 0$

Rewrite $1$ as $1^{2}$ .

$(- b + 6) ({(2 b)}^{2} - 1^{2}) = 0$

Factor.

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Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 b$ and $b = 1$ .

$(- b + 6) ((2 b + 1) (2 b - 1)) = 0$

Remove unnecessary parentheses.

$(- b + 6) (2 b + 1) (2 b - 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$- b + 6 = 0$

$2 b + 1 = 0$

$2 b - 1 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$- b + 6 = 0$

Subtract $6$ from both sides of the equation.

$- b = - 6$

Multiply each term in $- b = - 6$ by $- 1$

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Multiply each term in $- b = - 6$ by $- 1$ .

$(- b) \cdot - 1 = (- 6) \cdot - 1$

Multiply $(- b) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 b = (- 6) \cdot - 1$

Multiply $b$ by $1$ .

$b = (- 6) \cdot - 1$

Multiply $- 6$ by $- 1$ .

$b = 6$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$2 b + 1 = 0$

Subtract $1$ from both sides of the equation.

$2 b = - 1$

Divide each term by $2$ and simplify.

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Divide each term in $2 b = - 1$ by $2$ .

$\frac{2 b}{2} = \frac{- 1}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 b}{2} = \frac{- 1}{2}$

Divide $b$ by $1$ .

$b = \frac{- 1}{2}$

Move the negative in front of the fraction.

$b = - \frac{1}{2}$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$2 b - 1 = 0$

Add $1$ to both sides of the equation.

$2 b = 1$

Divide each term by $2$ and simplify.

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Divide each term in $2 b = 1$ by $2$ .

$\frac{2 b}{2} = \frac{1}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 b}{2} = \frac{1}{2}$

Divide $b$ by $1$ .

$b = \frac{1}{2}$

The final solution is all the values that make $(- b + 6) (2 b + 1) (2 b - 1) = 0$ true.

$b = 6, - \frac{1}{2}, \frac{1}{2}$

Exclude the solutions that do not make $\sqrt{6 - b} = 2 b \sqrt{6 - b}$ true.

$b = 6, \frac{1}{2}$

The result can be shown in multiple forms.

Exact Form:

$b = 6, \frac{1}{2}$

Decimal Form:

$b = 6, 0.5$

Solve for b square root of 6-b=2b square root of 6-b

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