Algebra Examples

$6 x = 2 \sqrt{24 x + 17} - 8$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 \sqrt{24 x + 17} - 8 = 6 x$

Step 2

Add $8$ to both sides of the equation.

$2 \sqrt{24 x + 17} = 6 x + 8$

Step 3

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

${(2 \sqrt{24 x + 17})}^{2} = {(6 x + 8)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{24 x + 17}$ as ${(24 x + 17)}^{\frac{1}{2}}$ .

${(2 {(24 x + 17)}^{\frac{1}{2}})}^{2} = {(6 x + 8)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(2 {(24 x + 17)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Apply the product rule to $2 {(24 x + 17)}^{\frac{1}{2}}$ .

$2^{2} {({(24 x + 17)}^{\frac{1}{2}})}^{2} = {(6 x + 8)}^{2}$

Step 4.2.1.2

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

$4 {({(24 x + 17)}^{\frac{1}{2}})}^{2} = {(6 x + 8)}^{2}$

Step 4.2.1.3

Multiply the exponents in ${({(24 x + 17)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.3.1

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

$4 {(24 x + 17)}^{\frac{1}{2} \cdot 2} = {(6 x + 8)}^{2}$

Step 4.2.1.3.2

Cancel the common factor of $2$ .

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Step 4.2.1.3.2.1

Cancel the common factor.

$4 {(24 x + 17)}^{\frac{1}{2} \cdot 2} = {(6 x + 8)}^{2}$

Step 4.2.1.3.2.2

Rewrite the expression.

$4 {(24 x + 17)}^{1} = {(6 x + 8)}^{2}$

Step 4.2.1.4

Simplify.

$4 (24 x + 17) = {(6 x + 8)}^{2}$

Step 4.2.1.5

Apply the distributive property.

$4 (24 x) + 4 \cdot 17 = {(6 x + 8)}^{2}$

Step 4.2.1.6

Multiply.

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Step 4.2.1.6.1

Multiply $24$ by $4$ .

$96 x + 4 \cdot 17 = {(6 x + 8)}^{2}$

Step 4.2.1.6.2

Multiply $4$ by $17$ .

$96 x + 68 = {(6 x + 8)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(6 x + 8)}^{2}$ .

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Step 4.3.1.1

Rewrite ${(6 x + 8)}^{2}$ as $(6 x + 8) (6 x + 8)$ .

$96 x + 68 = (6 x + 8) (6 x + 8)$

Step 4.3.1.2

Expand $(6 x + 8) (6 x + 8)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$96 x + 68 = 6 x (6 x + 8) + 8 (6 x + 8)$

Step 4.3.1.2.2

Apply the distributive property.

$96 x + 68 = 6 x (6 x) + 6 x \cdot 8 + 8 (6 x + 8)$

Step 4.3.1.2.3

Apply the distributive property.

$96 x + 68 = 6 x (6 x) + 6 x \cdot 8 + 8 (6 x) + 8 \cdot 8$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$96 x + 68 = 6 \cdot 6 x \cdot x + 6 x \cdot 8 + 8 (6 x) + 8 \cdot 8$

Step 4.3.1.3.1.2

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

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Step 4.3.1.3.1.2.1

Move $x$ .

$96 x + 68 = 6 \cdot 6 (x \cdot x) + 6 x \cdot 8 + 8 (6 x) + 8 \cdot 8$

Step 4.3.1.3.1.2.2

Multiply $x$ by $x$ .

$96 x + 68 = 6 \cdot 6 x^{2} + 6 x \cdot 8 + 8 (6 x) + 8 \cdot 8$

Step 4.3.1.3.1.3

Multiply $6$ by $6$ .

$96 x + 68 = 36 x^{2} + 6 x \cdot 8 + 8 (6 x) + 8 \cdot 8$

Step 4.3.1.3.1.4

Multiply $8$ by $6$ .

$96 x + 68 = 36 x^{2} + 48 x + 8 (6 x) + 8 \cdot 8$

Step 4.3.1.3.1.5

Multiply $6$ by $8$ .

$96 x + 68 = 36 x^{2} + 48 x + 48 x + 8 \cdot 8$

Step 4.3.1.3.1.6

Multiply $8$ by $8$ .

$96 x + 68 = 36 x^{2} + 48 x + 48 x + 64$

Step 4.3.1.3.2

Add $48 x$ and $48 x$ .

$96 x + 68 = 36 x^{2} + 96 x + 64$

Step 5

Solve for $x$ .

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Step 5.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$36 x^{2} + 96 x + 64 = 96 x + 68$

Step 5.2

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

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Step 5.2.1

Subtract $96 x$ from both sides of the equation.

$36 x^{2} + 96 x + 64 - 96 x = 68$

Step 5.2.2

Combine the opposite terms in $36 x^{2} + 96 x + 64 - 96 x$ .

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Step 5.2.2.1

Subtract $96 x$ from $96 x$ .

$36 x^{2} + 0 + 64 = 68$

Step 5.2.2.2

Add $36 x^{2}$ and $0$ .

$36 x^{2} + 64 = 68$

Step 5.3

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

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Step 5.3.1

Subtract $64$ from both sides of the equation.

$36 x^{2} = 68 - 64$

Step 5.3.2

Subtract $64$ from $68$ .

$36 x^{2} = 4$

Step 5.4

Divide each term in $36 x^{2} = 4$ by $36$ and simplify.

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Step 5.4.1

Divide each term in $36 x^{2} = 4$ by $36$ .

$\frac{36 x^{2}}{36} = \frac{4}{36}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $36$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{36 x^{2}}{36} = \frac{4}{36}$

Step 5.4.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{4}{36}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Cancel the common factor of $4$ and $36$ .

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Step 5.4.3.1.1

Factor $4$ out of $4$ .

$x^{2} = \frac{4 (1)}{36}$

Step 5.4.3.1.2

Cancel the common factors.

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Step 5.4.3.1.2.1

Factor $4$ out of $36$ .

$x^{2} = \frac{4 \cdot 1}{4 \cdot 9}$

Step 5.4.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{4 \cdot 1}{4 \cdot 9}$

Step 5.4.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{1}{9}$

Step 5.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{9}}$

Step 5.6

Simplify $\pm \sqrt{\frac{1}{9}}$ .

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Step 5.6.1

Rewrite $\sqrt{\frac{1}{9}}$ as $\frac{\sqrt{1}}{\sqrt{9}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{9}}$

Step 5.6.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{9}}$

Step 5.6.3

Simplify the denominator.

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Step 5.6.3.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \frac{1}{\sqrt{3^{2}}}$

Step 5.6.3.2

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

$x = \pm \frac{1}{3}$

Step 5.7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.7.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{1}{3}$

Step 5.7.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{1}{3}$

Step 5.7.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{1}{3}, - \frac{1}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}, - \frac{1}{3}$

Decimal Form:

$x = 0.\overline{3}, - 0.\overline{3}$