Solve for q square root of 3q+2 = square root of 5

$\sqrt{3 q} + 2 = \sqrt{5}$

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt{3 q} = \sqrt{5} - 2$

Step 2

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

${\sqrt{3 q}}^{2} = {(\sqrt{5} - 2)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 q}$ as ${(3 q)}^{\frac{1}{2}}$ .

${({(3 q)}^{\frac{1}{2}})}^{2} = {(\sqrt{5} - 2)}^{2}$

Simplify the left side.

Simplify ${({(3 q)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 q)}^{\frac{1}{2}})}^{2}$ .

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

${(3 q)}^{\frac{1}{2} \cdot 2} = {(\sqrt{5} - 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 q)}^{\frac{1}{2} \cdot 2} = {(\sqrt{5} - 2)}^{2}$

Rewrite the expression.

${(3 q)}^{1} = {(\sqrt{5} - 2)}^{2}$

Simplify.

$3 q = {(\sqrt{5} - 2)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{5} - 2)}^{2}$ .

Rewrite ${(\sqrt{5} - 2)}^{2}$ as $(\sqrt{5} - 2) (\sqrt{5} - 2)$ .

$3 q = (\sqrt{5} - 2) (\sqrt{5} - 2)$

Expand $(\sqrt{5} - 2) (\sqrt{5} - 2)$ using the FOIL Method.

Apply the distributive property.

$3 q = \sqrt{5} (\sqrt{5} - 2) - 2 (\sqrt{5} - 2)$

Apply the distributive property.

$3 q = \sqrt{5} \sqrt{5} + \sqrt{5} \cdot - 2 - 2 (\sqrt{5} - 2)$

Apply the distributive property.

$3 q = \sqrt{5} \sqrt{5} + \sqrt{5} \cdot - 2 - 2 \sqrt{5} - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$3 q = \sqrt{5 \cdot 5} + \sqrt{5} \cdot - 2 - 2 \sqrt{5} - 2 \cdot - 2$

Multiply $5$ by $5$ .

$3 q = \sqrt{25} + \sqrt{5} \cdot - 2 - 2 \sqrt{5} - 2 \cdot - 2$

Rewrite $25$ as $5^{2}$ .

$3 q = \sqrt{5^{2}} + \sqrt{5} \cdot - 2 - 2 \sqrt{5} - 2 \cdot - 2$

Pull terms out from under the radical, assuming positive real numbers.

$3 q = 5 + \sqrt{5} \cdot - 2 - 2 \sqrt{5} - 2 \cdot - 2$

Move $- 2$ to the left of $\sqrt{5}$ .

$3 q = 5 - 2 \cdot \sqrt{5} - 2 \sqrt{5} - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$3 q = 5 - 2 \sqrt{5} - 2 \sqrt{5} + 4$

Add $5$ and $4$ .

$3 q = 9 - 2 \sqrt{5} - 2 \sqrt{5}$

Subtract $2 \sqrt{5}$ from $- 2 \sqrt{5}$ .

$3 q = 9 - 4 \sqrt{5}$

Step 4

Divide each term in $3 q = 9 - 4 \sqrt{5}$ by $3$ and simplify.

Divide each term in $3 q = 9 - 4 \sqrt{5}$ by $3$ .

$\frac{3 q}{3} = \frac{9}{3} + \frac{- 4 \sqrt{5}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 q}{3} = \frac{9}{3} + \frac{- 4 \sqrt{5}}{3}$

Divide $q$ by $1$ .

$q = \frac{9}{3} + \frac{- 4 \sqrt{5}}{3}$

Simplify the right side.

Simplify each term.

Divide $9$ by $3$ .

$q = 3 + \frac{- 4 \sqrt{5}}{3}$

Move the negative in front of the fraction.

$q = 3 - \frac{4 \sqrt{5}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$q = 3 - \frac{4 \sqrt{5}}{3}$

Decimal Form:

$q = 0.01857603 \dots$

Solve for q square root of 3q+2 = square root of 5

Related Questions

Support LCMGCF.com by disabling your adblocker.