Algebra Examples

$\sqrt{12 x - 23} = \sqrt{3 x - 11} + \sqrt{3 x}$

Step 1

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

${\sqrt{12 x - 23}}^{2} = {(\sqrt{3 x - 11} + \sqrt{3 x})}^{2}$

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 $\sqrt{12 x - 23}$ as ${(12 x - 23)}^{\frac{1}{2}}$ .

${({(12 x - 23)}^{\frac{1}{2}})}^{2} = {(\sqrt{3 x - 11} + \sqrt{3 x})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(12 x - 23)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(12 x - 23)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(12 x - 23)}^{\frac{1}{2} \cdot 2} = {(\sqrt{3 x - 11} + \sqrt{3 x})}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(12 x - 23)}^{\frac{1}{2} \cdot 2} = {(\sqrt{3 x - 11} + \sqrt{3 x})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(12 x - 23)}^{1} = {(\sqrt{3 x - 11} + \sqrt{3 x})}^{2}$

Step 2.2.1.2

Simplify.

$12 x - 23 = {(\sqrt{3 x - 11} + \sqrt{3 x})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(\sqrt{3 x - 11} + \sqrt{3 x})}^{2}$ .

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Step 2.3.1.1

Rewrite ${(\sqrt{3 x - 11} + \sqrt{3 x})}^{2}$ as $(\sqrt{3 x - 11} + \sqrt{3 x}) (\sqrt{3 x - 11} + \sqrt{3 x})$ .

$12 x - 23 = (\sqrt{3 x - 11} + \sqrt{3 x}) (\sqrt{3 x - 11} + \sqrt{3 x})$

Step 2.3.1.2

Expand $(\sqrt{3 x - 11} + \sqrt{3 x}) (\sqrt{3 x - 11} + \sqrt{3 x})$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$12 x - 23 = \sqrt{3 x - 11} (\sqrt{3 x - 11} + \sqrt{3 x}) + \sqrt{3 x} (\sqrt{3 x - 11} + \sqrt{3 x})$

Step 2.3.1.2.2

Apply the distributive property.

$12 x - 23 = \sqrt{3 x - 11} \sqrt{3 x - 11} + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} (\sqrt{3 x - 11} + \sqrt{3 x})$

Step 2.3.1.2.3

Apply the distributive property.

$12 x - 23 = \sqrt{3 x - 11} \sqrt{3 x - 11} + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $\sqrt{3 x - 11} \sqrt{3 x - 11}$ .

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Step 2.3.1.3.1.1.1

Raise $\sqrt{3 x - 11}$ to the power of $1$ .

$12 x - 23 = {\sqrt{3 x - 11}}^{1} \sqrt{3 x - 11} + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.1.2

Raise $\sqrt{3 x - 11}$ to the power of $1$ .

$12 x - 23 = {\sqrt{3 x - 11}}^{1} {\sqrt{3 x - 11}}^{1} + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x - 23 = {\sqrt{3 x - 11}}^{1 + 1} + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.1.4

Add $1$ and $1$ .

$12 x - 23 = {\sqrt{3 x - 11}}^{2} + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.2

Rewrite ${\sqrt{3 x - 11}}^{2}$ as $3 x - 11$ .

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Step 2.3.1.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 11}$ as ${(3 x - 11)}^{\frac{1}{2}}$ .

$12 x - 23 = {({(3 x - 11)}^{\frac{1}{2}})}^{2} + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.2.2

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

$12 x - 23 = {(3 x - 11)}^{\frac{1}{2} \cdot 2} + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.2.3

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

$12 x - 23 = {(3 x - 11)}^{\frac{2}{2}} + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.2.4.1

Cancel the common factor.

$12 x - 23 = {(3 x - 11)}^{\frac{2}{2}} + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.2.4.2

Rewrite the expression.

$12 x - 23 = {(3 x - 11)}^{1} + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.2.5

Simplify.

$12 x - 23 = 3 x - 11 + \sqrt{3 x - 11} \sqrt{3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.3

Combine using the product rule for radicals.

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x} \sqrt{3 x - 11} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.4

Combine using the product rule for radicals.

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)} + \sqrt{3 x} \sqrt{3 x}$

Step 2.3.1.3.1.5

Multiply $\sqrt{3 x} \sqrt{3 x}$ .

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Step 2.3.1.3.1.5.1

Raise $\sqrt{3 x}$ to the power of $1$ .

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)} + {\sqrt{3 x}}^{1} \sqrt{3 x}$

Step 2.3.1.3.1.5.2

Raise $\sqrt{3 x}$ to the power of $1$ .

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)} + {\sqrt{3 x}}^{1} {\sqrt{3 x}}^{1}$

Step 2.3.1.3.1.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)} + {\sqrt{3 x}}^{1 + 1}$

Step 2.3.1.3.1.5.4

Add $1$ and $1$ .

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)} + {\sqrt{3 x}}^{2}$

Step 2.3.1.3.1.6

Rewrite ${\sqrt{3 x}}^{2}$ as $3 x$ .

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Step 2.3.1.3.1.6.1

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

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)} + {({(3 x)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.3.1.6.2

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

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)} + {(3 x)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3.1.6.3

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

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)} + {(3 x)}^{\frac{2}{2}}$

Step 2.3.1.3.1.6.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.6.4.1

Cancel the common factor.

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)} + {(3 x)}^{\frac{2}{2}}$

Step 2.3.1.3.1.6.4.2

Rewrite the expression.

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)} + {(3 x)}^{1}$

Step 2.3.1.3.1.6.5

Simplify.

$12 x - 23 = 3 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)} + 3 x$

Step 2.3.1.3.2

Add $3 x$ and $3 x$ .

$12 x - 23 = 6 x - 11 + \sqrt{(3 x - 11) \cdot 3 x} + \sqrt{3 x (3 x - 11)}$

Step 2.3.1.3.3

Add $\sqrt{(3 x - 11) \cdot 3 x}$ and $\sqrt{3 x (3 x - 11)}$ .

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Step 2.3.1.3.3.1

Move $3 x - 11$ .

$12 x - 23 = 6 x - 11 + \sqrt{3 x (3 x - 11)} + \sqrt{3 x (3 x - 11)}$

Step 2.3.1.3.3.2

Add $\sqrt{3 x (3 x - 11)}$ and $\sqrt{3 x (3 x - 11)}$ .

$12 x - 23 = 6 x - 11 + 2 \sqrt{3 x (3 x - 11)}$

Step 3

Solve for $2 \sqrt{3 x (3 x - 11)}$ .

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Step 3.1

Rewrite the equation as $6 x - 11 + 2 \sqrt{3 x (3 x - 11)} = 12 x - 23$ .

$6 x - 11 + 2 \sqrt{3 x (3 x - 11)} = 12 x - 23$

Step 3.2

Move all terms not containing $2 \sqrt{3 x (3 x - 11)}$ to the right side of the equation.

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Step 3.2.1

Subtract $6 x$ from both sides of the equation.

$- 11 + 2 \sqrt{3 x (3 x - 11)} = 12 x - 23 - 6 x$

Step 3.2.2

Add $11$ to both sides of the equation.

$2 \sqrt{3 x (3 x - 11)} = 12 x - 23 - 6 x + 11$

Step 3.2.3

Subtract $6 x$ from $12 x$ .

$2 \sqrt{3 x (3 x - 11)} = 6 x - 23 + 11$

Step 3.2.4

Add $- 23$ and $11$ .

$2 \sqrt{3 x (3 x - 11)} = 6 x - 12$

Step 4

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

${(2 \sqrt{3 x (3 x - 11)})}^{2} = {(6 x - 12)}^{2}$

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 $\sqrt{3 x (3 x - 11)}$ as ${(3 x (3 x - 11))}^{\frac{1}{2}}$ .

${(2 {(3 x (3 x - 11))}^{\frac{1}{2}})}^{2} = {(6 x - 12)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(2 {(3 x (3 x - 11))}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Simplify by multiplying through.

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Step 5.2.1.1.1

Apply the distributive property.

${(2 {(3 x (3 x) + 3 x \cdot - 11)}^{\frac{1}{2}})}^{2} = {(6 x - 12)}^{2}$

Step 5.2.1.1.2

Simplify the expression.

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Step 5.2.1.1.2.1

Rewrite using the commutative property of multiplication.

${(2 {(3 \cdot 3 x \cdot x + 3 x \cdot - 11)}^{\frac{1}{2}})}^{2} = {(6 x - 12)}^{2}$

Step 5.2.1.1.2.2

Multiply $- 11$ by $3$ .

${(2 {(3 \cdot 3 x \cdot x - 33 x)}^{\frac{1}{2}})}^{2} = {(6 x - 12)}^{2}$

Step 5.2.1.2

Simplify each term.

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Step 5.2.1.2.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.1.2.1.1

Move $x$ .

${(2 {(3 \cdot 3 (x \cdot x) - 33 x)}^{\frac{1}{2}})}^{2} = {(6 x - 12)}^{2}$

Step 5.2.1.2.1.2

Multiply $x$ by $x$ .

${(2 {(3 \cdot 3 x^{2} - 33 x)}^{\frac{1}{2}})}^{2} = {(6 x - 12)}^{2}$

Step 5.2.1.2.2

Multiply $3$ by $3$ .

${(2 {(9 x^{2} - 33 x)}^{\frac{1}{2}})}^{2} = {(6 x - 12)}^{2}$

Step 5.2.1.3

Simplify the expression.

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Step 5.2.1.3.1

Apply the product rule to $2 {(9 x^{2} - 33 x)}^{\frac{1}{2}}$ .

$2^{2} {({(9 x^{2} - 33 x)}^{\frac{1}{2}})}^{2} = {(6 x - 12)}^{2}$

Step 5.2.1.3.2

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

$4 {({(9 x^{2} - 33 x)}^{\frac{1}{2}})}^{2} = {(6 x - 12)}^{2}$

Step 5.2.1.3.3

Multiply the exponents in ${({(9 x^{2} - 33 x)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.3.3.1

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

$4 {(9 x^{2} - 33 x)}^{\frac{1}{2} \cdot 2} = {(6 x - 12)}^{2}$

Step 5.2.1.3.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.3.2.1

Cancel the common factor.

$4 {(9 x^{2} - 33 x)}^{\frac{1}{2} \cdot 2} = {(6 x - 12)}^{2}$

Step 5.2.1.3.3.2.2

Rewrite the expression.

$4 {(9 x^{2} - 33 x)}^{1} = {(6 x - 12)}^{2}$

Step 5.2.1.4

Simplify.

$4 (9 x^{2} - 33 x) = {(6 x - 12)}^{2}$

Step 5.2.1.5

Apply the distributive property.

$4 (9 x^{2}) + 4 (- 33 x) = {(6 x - 12)}^{2}$

Step 5.2.1.6

Multiply.

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Step 5.2.1.6.1

Multiply $9$ by $4$ .

$36 x^{2} + 4 (- 33 x) = {(6 x - 12)}^{2}$

Step 5.2.1.6.2

Multiply $- 33$ by $4$ .

$36 x^{2} - 132 x = {(6 x - 12)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify ${(6 x - 12)}^{2}$ .

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Step 5.3.1.1

Rewrite ${(6 x - 12)}^{2}$ as $(6 x - 12) (6 x - 12)$ .

$36 x^{2} - 132 x = (6 x - 12) (6 x - 12)$

Step 5.3.1.2

Expand $(6 x - 12) (6 x - 12)$ using the FOIL Method.

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Step 5.3.1.2.1

Apply the distributive property.

$36 x^{2} - 132 x = 6 x (6 x - 12) - 12 (6 x - 12)$

Step 5.3.1.2.2

Apply the distributive property.

$36 x^{2} - 132 x = 6 x (6 x) + 6 x \cdot - 12 - 12 (6 x - 12)$

Step 5.3.1.2.3

Apply the distributive property.

$36 x^{2} - 132 x = 6 x (6 x) + 6 x \cdot - 12 - 12 (6 x) - 12 \cdot - 12$

Step 5.3.1.3

Simplify and combine like terms.

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Step 5.3.1.3.1

Simplify each term.

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Step 5.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$36 x^{2} - 132 x = 6 \cdot 6 x \cdot x + 6 x \cdot - 12 - 12 (6 x) - 12 \cdot - 12$

Step 5.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.1.3.1.2.1

Move $x$ .

$36 x^{2} - 132 x = 6 \cdot 6 (x \cdot x) + 6 x \cdot - 12 - 12 (6 x) - 12 \cdot - 12$

Step 5.3.1.3.1.2.2

Multiply $x$ by $x$ .

$36 x^{2} - 132 x = 6 \cdot 6 x^{2} + 6 x \cdot - 12 - 12 (6 x) - 12 \cdot - 12$

Step 5.3.1.3.1.3

Multiply $6$ by $6$ .

$36 x^{2} - 132 x = 36 x^{2} + 6 x \cdot - 12 - 12 (6 x) - 12 \cdot - 12$

Step 5.3.1.3.1.4

Multiply $- 12$ by $6$ .

$36 x^{2} - 132 x = 36 x^{2} - 72 x - 12 (6 x) - 12 \cdot - 12$

Step 5.3.1.3.1.5

Multiply $6$ by $- 12$ .

$36 x^{2} - 132 x = 36 x^{2} - 72 x - 72 x - 12 \cdot - 12$

Step 5.3.1.3.1.6

Multiply $- 12$ by $- 12$ .

$36 x^{2} - 132 x = 36 x^{2} - 72 x - 72 x + 144$

Step 5.3.1.3.2

Subtract $72 x$ from $- 72 x$ .

$36 x^{2} - 132 x = 36 x^{2} - 144 x + 144$

Step 6

Solve for $x$ .

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Step 6.1

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

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Step 6.1.1

Subtract $36 x^{2}$ from both sides of the equation.

$36 x^{2} - 132 x - 36 x^{2} = - 144 x + 144$

Step 6.1.2

Add $144 x$ to both sides of the equation.

$36 x^{2} - 132 x - 36 x^{2} + 144 x = 144$

Step 6.1.3

Combine the opposite terms in $36 x^{2} - 132 x - 36 x^{2} + 144 x$ .

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Step 6.1.3.1

Subtract $36 x^{2}$ from $36 x^{2}$ .

$- 132 x + 0 + 144 x = 144$

Step 6.1.3.2

Add $- 132 x$ and $0$ .

$- 132 x + 144 x = 144$

Step 6.1.4

Add $- 132 x$ and $144 x$ .

$12 x = 144$

Step 6.2

Divide each term in $12 x = 144$ by $12$ and simplify.

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Step 6.2.1

Divide each term in $12 x = 144$ by $12$ .

$\frac{12 x}{12} = \frac{144}{12}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $12$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{144}{12}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{144}{12}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $144$ by $12$ .

$x = 12$

Step 7