Algebra Examples

$2 \sqrt{10 + 3 x} = \sqrt{7 + x}$

Step 1

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

${(2 \sqrt{10 + 3 x})}^{2} = {\sqrt{7 + x}}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 + 3 x}$ as ${(10 + 3 x)}^{\frac{1}{2}}$ .

${(2 {(10 + 3 x)}^{\frac{1}{2}})}^{2} = {\sqrt{7 + x}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(2 {(10 + 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply the product rule to $2 {(10 + 3 x)}^{\frac{1}{2}}$ .

$2^{2} {({(10 + 3 x)}^{\frac{1}{2}})}^{2} = {\sqrt{7 + x}}^{2}$

Step 2.2.1.2

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

$4 {({(10 + 3 x)}^{\frac{1}{2}})}^{2} = {\sqrt{7 + x}}^{2}$

Step 2.2.1.3

Multiply the exponents in ${({(10 + 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.3.1

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

$4 {(10 + 3 x)}^{\frac{1}{2} \cdot 2} = {\sqrt{7 + x}}^{2}$

Step 2.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.2.1.3.2.1

Cancel the common factor.

$4 {(10 + 3 x)}^{\frac{1}{2} \cdot 2} = {\sqrt{7 + x}}^{2}$

Step 2.2.1.3.2.2

Rewrite the expression.

$4 {(10 + 3 x)}^{1} = {\sqrt{7 + x}}^{2}$

Step 2.2.1.4

Simplify.

$4 (10 + 3 x) = {\sqrt{7 + x}}^{2}$

Step 2.2.1.5

Apply the distributive property.

$4 \cdot 10 + 4 (3 x) = {\sqrt{7 + x}}^{2}$

Step 2.2.1.6

Multiply.

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Step 2.2.1.6.1

Multiply $4$ by $10$ .

$40 + 4 (3 x) = {\sqrt{7 + x}}^{2}$

Step 2.2.1.6.2

Multiply $3$ by $4$ .

$40 + 12 x = {\sqrt{7 + x}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt{7 + x}}^{2}$ as $7 + x$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 + x}$ as ${(7 + x)}^{\frac{1}{2}}$ .

$40 + 12 x = {({(7 + x)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2

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

$40 + 12 x = {(7 + x)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3

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

$40 + 12 x = {(7 + x)}^{\frac{2}{2}}$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

$40 + 12 x = {(7 + x)}^{\frac{2}{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$40 + 12 x = {(7 + x)}^{1}$

Step 2.3.1.5

Simplify.

$40 + 12 x = 7 + x$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $x$ from both sides of the equation.

$40 + 12 x - x = 7$

Step 3.1.2

Subtract $x$ from $12 x$ .

$40 + 11 x = 7$

Step 3.2

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

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Step 3.2.1

Subtract $40$ from both sides of the equation.

$11 x = 7 - 40$

Step 3.2.2

Subtract $40$ from $7$ .

$11 x = - 33$

Step 3.3

Divide each term in $11 x = - 33$ by $11$ and simplify.

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Step 3.3.1

Divide each term in $11 x = - 33$ by $11$ .

$\frac{11 x}{11} = \frac{- 33}{11}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $11$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{- 33}{11}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 33}{11}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 33$ by $11$ .

$x = - 3$