Algebra Examples

$2 = \frac{1 + \sqrt{29 - 4 x}}{- 3}$

Step 1

Rewrite the equation as $\frac{1 + \sqrt{29 - 4 x}}{- 3} = 2$ .

$\frac{1 + \sqrt{29 - 4 x}}{- 3} = 2$

Step 2

Multiply both sides by $- 3$ .

$\frac{1 + \sqrt{29 - 4 x}}{- 3} \cdot - 3 = 2 \cdot - 3$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{1 + \sqrt{29 - 4 x}}{- 3} \cdot - 3$ .

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Step 3.1.1.1

Simplify terms.

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Step 3.1.1.1.1

Cancel the common factor of $3$ .

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Step 3.1.1.1.1.1

Factor $3$ out of $- 3$ .

$\frac{1 + \sqrt{29 - 4 x}}{3 (- 1)} \cdot - 3 = 2 \cdot - 3$

Step 3.1.1.1.1.2

Factor $3$ out of $- 3$ .

$\frac{1 + \sqrt{29 - 4 x}}{3 \cdot - 1} \cdot (3 \cdot - 1) = 2 \cdot - 3$

Step 3.1.1.1.1.3

Cancel the common factor.

$\frac{1 + \sqrt{29 - 4 x}}{3 \cdot - 1} \cdot (3 \cdot - 1) = 2 \cdot - 3$

Step 3.1.1.1.1.4

Rewrite the expression.

$\frac{1 + \sqrt{29 - 4 x}}{- 1} \cdot - 1 = 2 \cdot - 3$

Step 3.1.1.1.2

Combine $\frac{1 + \sqrt{29 - 4 x}}{- 1}$ and $- 1$ .

$\frac{(1 + \sqrt{29 - 4 x}) \cdot - 1}{- 1} = 2 \cdot - 3$

Step 3.1.1.1.3

Simplify the expression.

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Step 3.1.1.1.3.1

Move the negative one from the denominator of $\frac{(1 + \sqrt{29 - 4 x}) \cdot - 1}{- 1}$ .

$- 1 \cdot ((1 + \sqrt{29 - 4 x}) \cdot - 1) = 2 \cdot - 3$

Step 3.1.1.1.3.2

Rewrite $- 1 \cdot ((1 + \sqrt{29 - 4 x}) \cdot - 1)$ as $- ((1 + \sqrt{29 - 4 x}) \cdot - 1)$ .

$- ((1 + \sqrt{29 - 4 x}) \cdot - 1) = 2 \cdot - 3$

Step 3.1.1.1.4

Apply the distributive property.

$- (1 \cdot - 1 + \sqrt{29 - 4 x} \cdot - 1) = 2 \cdot - 3$

Step 3.1.1.1.5

Simplify the expression.

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Step 3.1.1.1.5.1

Multiply $- 1$ by $1$ .

$- (- 1 + \sqrt{29 - 4 x} \cdot - 1) = 2 \cdot - 3$

Step 3.1.1.1.5.2

Move $- 1$ to the left of $\sqrt{29 - 4 x}$ .

$- (- 1 - 1 \cdot \sqrt{29 - 4 x}) = 2 \cdot - 3$

Step 3.1.1.2

Rewrite $- 1 \sqrt{29 - 4 x}$ as $- \sqrt{29 - 4 x}$ .

$- (- 1 - \sqrt{29 - 4 x}) = 2 \cdot - 3$

Step 3.1.1.3

Simplify by multiplying through.

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Step 3.1.1.3.1

Apply the distributive property.

$- - 1 - - \sqrt{29 - 4 x} = 2 \cdot - 3$

Step 3.1.1.3.2

Multiply $- 1$ by $- 1$ .

$1 - - \sqrt{29 - 4 x} = 2 \cdot - 3$

Step 3.1.1.4

Multiply $- - \sqrt{29 - 4 x}$ .

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Step 3.1.1.4.1

Multiply $- 1$ by $- 1$ .

$1 + 1 \sqrt{29 - 4 x} = 2 \cdot - 3$

Step 3.1.1.4.2

Multiply $\sqrt{29 - 4 x}$ by $1$ .

$1 + \sqrt{29 - 4 x} = 2 \cdot - 3$

Step 3.1.1.5

Reorder $29$ and $- 4 x$ .

$1 + \sqrt{- 4 x + 29} = 2 \cdot - 3$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $2$ by $- 3$ .

$1 + \sqrt{- 4 x + 29} = - 6$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms not containing $\sqrt{- 4 x + 29}$ to the right side of the equation.

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Step 4.1.1

Subtract $1$ from both sides of the equation.

$\sqrt{- 4 x + 29} = - 6 - 1$

Step 4.1.2

Subtract $1$ from $- 6$ .

$\sqrt{- 4 x + 29} = - 7$

Step 4.2

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

${\sqrt{- 4 x + 29}}^{2} = {(- 7)}^{2}$

Step 4.3

Simplify each side of the equation.

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Step 4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 4 x + 29}$ as ${(- 4 x + 29)}^{\frac{1}{2}}$ .

${({(- 4 x + 29)}^{\frac{1}{2}})}^{2} = {(- 7)}^{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Simplify ${({(- 4 x + 29)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.1.1

Multiply the exponents in ${({(- 4 x + 29)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.1.1.1

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

${(- 4 x + 29)}^{\frac{1}{2} \cdot 2} = {(- 7)}^{2}$

Step 4.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.1.2.1

Cancel the common factor.

${(- 4 x + 29)}^{\frac{1}{2} \cdot 2} = {(- 7)}^{2}$

Step 4.3.2.1.1.2.2

Rewrite the expression.

${(- 4 x + 29)}^{1} = {(- 7)}^{2}$

Step 4.3.2.1.2

Simplify.

$- 4 x + 29 = {(- 7)}^{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Raise $- 7$ to the power of $2$ .

$- 4 x + 29 = 49$

Step 4.4

Solve for $x$ .

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Step 4.4.1

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

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Step 4.4.1.1

Subtract $29$ from both sides of the equation.

$- 4 x = 49 - 29$

Step 4.4.1.2

Subtract $29$ from $49$ .

$- 4 x = 20$

Step 4.4.2

Divide each term in $- 4 x = 20$ by $- 4$ and simplify.

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Step 4.4.2.1

Divide each term in $- 4 x = 20$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{20}{- 4}$

Step 4.4.2.2

Simplify the left side.

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Step 4.4.2.2.1

Cancel the common factor of $- 4$ .

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Step 4.4.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{20}{- 4}$

Step 4.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{20}{- 4}$

Step 4.4.2.3

Simplify the right side.

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Step 4.4.2.3.1

Divide $20$ by $- 4$ .

$x = - 5$

Step 5

Exclude the solutions that do not make $2 = \frac{1 + \sqrt{29 - 4 x}}{- 3}$ true.

$No solution$