Solve for q square root of 2q-1 = square root of 3q-6

$\sqrt{2 q - 1} = \sqrt{3 q - 6}$

Step 1

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

${\sqrt{2 q - 1}}^{2} = {\sqrt{3 q - 6}}^{2}$

Step 2

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$2 q - 1 = {\sqrt{3 q - 6}}^{2}$

Simplify the right side.

Simplify ${\sqrt{3 q - 6}}^{2}$ .

Factor $3$ out of $3 q - 6$ .

Factor $3$ out of $3 q$ .

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

Factor $3$ out of $- 6$ .

$2 q - 1 = {\sqrt{3 q + 3 \cdot - 2}}^{2}$

Factor $3$ out of $3 q + 3 \cdot - 2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$2 q - 1 = 3 (q - 2)$

Apply the distributive property.

$2 q - 1 = 3 q + 3 \cdot - 2$

Multiply $3$ by $- 2$ .

$2 q - 1 = 3 q - 6$

Step 3

Solve for $q$ .

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

Subtract $3 q$ from both sides of the equation.

$2 q - 1 - 3 q = - 6$

Subtract $3 q$ from $2 q$ .

$- q - 1 = - 6$

Move all terms not containing $q$ to the right side of the equation.

Add $1$ to both sides of the equation.

$- q = - 6 + 1$

Add $- 6$ and $1$ .

$- q = - 5$

Divide each term in $- q = - 5$ by $- 1$ and simplify.

Divide each term in $- q = - 5$ by $- 1$ .

$\frac{- q}{- 1} = \frac{- 5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{q}{1} = \frac{- 5}{- 1}$

Divide $q$ by $1$ .

$q = \frac{- 5}{- 1}$

Simplify the right side.

Divide $- 5$ by $- 1$ .

$q = 5$

Solve for q square root of 2q-1 = square root of 3q-6

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