Algebra Examples

$\sqrt{2 m - 6} = \sqrt{3 m - 14}$

Step 1

Subtract $\sqrt{3 m - 14}$ from both sides of the equation.

$\sqrt{2 m - 6} - \sqrt{3 m - 14} = 0$

Step 2

Factor $2$ out of $2 m - 6$ .

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Step 2.1

Factor $2$ out of $2 m$ .

$\sqrt{2 (m) - 6} - \sqrt{3 m - 14} = 0$

Step 2.2

Factor $2$ out of $- 6$ .

$\sqrt{2 m + 2 \cdot - 3} - \sqrt{3 m - 14} = 0$

Step 2.3

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

$\sqrt{2 (m - 3)} - \sqrt{3 m - 14} = 0$

Step 3

Add $\sqrt{3 m - 14}$ to both sides of the equation.

$\sqrt{2 (m - 3)} = \sqrt{3 m - 14}$

Step 4

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

${\sqrt{2 (m - 3)}}^{2} = {\sqrt{3 m - 14}}^{2}$

Step 5

Simplify each side of the equation.

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Step 5.1

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

${({(2 (m - 3))}^{\frac{1}{2}})}^{2} = {\sqrt{3 m - 14}}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

${(2 (m - 3))}^{\frac{1}{2} \cdot 2} = {\sqrt{3 m - 14}}^{2}$

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

${(2 (m - 3))}^{\frac{1}{2} \cdot 2} = {\sqrt{3 m - 14}}^{2}$

Step 5.2.1.1.2.2

Rewrite the expression.

${(2 (m - 3))}^{1} = {\sqrt{3 m - 14}}^{2}$

Step 5.2.1.2

Apply the distributive property.

${(2 m + 2 \cdot - 3)}^{1} = {\sqrt{3 m - 14}}^{2}$

Step 5.2.1.3

Multiply.

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Step 5.2.1.3.1

Multiply $2$ by $- 3$ .

${(2 m - 6)}^{1} = {\sqrt{3 m - 14}}^{2}$

Step 5.2.1.3.2

Simplify.

$2 m - 6 = {\sqrt{3 m - 14}}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Rewrite ${\sqrt{3 m - 14}}^{2}$ as $3 m - 14$ .

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Step 5.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 m - 14}$ as ${(3 m - 14)}^{\frac{1}{2}}$ .

$2 m - 6 = {({(3 m - 14)}^{\frac{1}{2}})}^{2}$

Step 5.3.1.2

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

$2 m - 6 = {(3 m - 14)}^{\frac{1}{2} \cdot 2}$

Step 5.3.1.3

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

$2 m - 6 = {(3 m - 14)}^{\frac{2}{2}}$

Step 5.3.1.4

Cancel the common factor of $2$ .

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Step 5.3.1.4.1

Cancel the common factor.

$2 m - 6 = {(3 m - 14)}^{\frac{2}{2}}$

Step 5.3.1.4.2

Rewrite the expression.

$2 m - 6 = {(3 m - 14)}^{1}$

Step 5.3.1.5

Simplify.

$2 m - 6 = 3 m - 14$

Step 6

Solve for $m$ .

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Step 6.1

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

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Step 6.1.1

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

$2 m - 6 - 3 m = - 14$

Step 6.1.2

Subtract $3 m$ from $2 m$ .

$- m - 6 = - 14$

Step 6.2

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

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Step 6.2.1

Add $6$ to both sides of the equation.

$- m = - 14 + 6$

Step 6.2.2

Add $- 14$ and $6$ .

$- m = - 8$

Step 6.3

Divide each term in $- m = - 8$ by $- 1$ and simplify.

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Step 6.3.1

Divide each term in $- m = - 8$ by $- 1$ .

$\frac{- m}{- 1} = \frac{- 8}{- 1}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Dividing two negative values results in a positive value.

$\frac{m}{1} = \frac{- 8}{- 1}$

Step 6.3.2.2

Divide $m$ by $1$ .

$m = \frac{- 8}{- 1}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $- 8$ by $- 1$ .

$m = 8$