Solve for c square root of 2c+1- square root of 4c=0

$\sqrt{2 c + 1} - \sqrt{4 c} = 0$

Step 1

Solve for $\sqrt{2 c + 1}$ .

Simplify each term.

Rewrite $4$ as $2^{2}$ .

$\sqrt{2 c + 1} - \sqrt{2^{2} c} = 0$

Pull terms out from under the radical.

$\sqrt{2 c + 1} - (2 \sqrt{c}) = 0$

Multiply $2$ by $- 1$ .

$\sqrt{2 c + 1} - 2 \sqrt{c} = 0$

Add $2 \sqrt{c}$ to both sides of the equation.

$\sqrt{2 c + 1} = 2 \sqrt{c}$

Step 2

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

${\sqrt{2 c + 1}}^{2} = {(2 \sqrt{c})}^{2}$

Step 3

Simplify each side of the equation.

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

${({(2 c + 1)}^{\frac{1}{2}})}^{2} = {(2 \sqrt{c})}^{2}$

Simplify the left side.

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

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

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

${(2 c + 1)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{c})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 c + 1)}^{\frac{1}{2} \cdot 2} = {(2 \sqrt{c})}^{2}$

Rewrite the expression.

${(2 c + 1)}^{1} = {(2 \sqrt{c})}^{2}$

Simplify.

$2 c + 1 = {(2 \sqrt{c})}^{2}$

Simplify the right side.

Simplify ${(2 \sqrt{c})}^{2}$ .

Simplify the expression.

Apply the product rule to $2 \sqrt{c}$ .

$2 c + 1 = 2^{2} {\sqrt{c}}^{2}$

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

$2 c + 1 = 4 {\sqrt{c}}^{2}$

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

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

$2 c + 1 = 4 {(c^{\frac{1}{2}})}^{2}$

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

$2 c + 1 = 4 c^{\frac{1}{2} \cdot 2}$

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

$2 c + 1 = 4 c^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 c + 1 = 4 c^{\frac{2}{2}}$

Rewrite the expression.

$2 c + 1 = 4 c^{1}$

Simplify.

$2 c + 1 = 4 c$

Step 4

Solve for $c$ .

Move all terms containing $c$ to the left side of the equation.

Subtract $4 c$ from both sides of the equation.

$2 c + 1 - 4 c = 0$

Subtract $4 c$ from $2 c$ .

$- 2 c + 1 = 0$

Subtract $1$ from both sides of the equation.

$- 2 c = - 1$

Divide each term in $- 2 c = - 1$ by $- 2$ and simplify.

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $c$ by $1$ .

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

Simplify the right side.

Dividing two negative values results in a positive value.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$c = 0.5$

Solve for c square root of 2c+1- square root of 4c=0

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