Pre-Algebra Examples

$5 \sqrt{5 x + 29} = x + 3$

Step 1

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

${(5 \sqrt{5 x + 29})}^{2} = {(x + 3)}^{2}$

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 $\sqrt{5 x + 29}$ as ${(5 x + 29)}^{\frac{1}{2}}$ .

${(5 {(5 x + 29)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(5 {(5 x + 29)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply the product rule to $5 {(5 x + 29)}^{\frac{1}{2}}$ .

$5^{2} {({(5 x + 29)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Step 2.2.1.2

Raise $5$ to the power of $2$ .

$25 {({(5 x + 29)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Step 2.2.1.3

Multiply the exponents in ${({(5 x + 29)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.3.1

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

$25 {(5 x + 29)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

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.

$25 {(5 x + 29)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 2.2.1.3.2.2

Rewrite the expression.

$25 {(5 x + 29)}^{1} = {(x + 3)}^{2}$

Step 2.2.1.4

Simplify.

$25 (5 x + 29) = {(x + 3)}^{2}$

Step 2.2.1.5

Apply the distributive property.

$25 (5 x) + 25 \cdot 29 = {(x + 3)}^{2}$

Step 2.2.1.6

Multiply.

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Step 2.2.1.6.1

Multiply $5$ by $25$ .

$125 x + 25 \cdot 29 = {(x + 3)}^{2}$

Step 2.2.1.6.2

Multiply $25$ by $29$ .

$125 x + 725 = {(x + 3)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(x + 3)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

$125 x + 725 = (x + 3) (x + 3)$

Step 2.3.1.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$125 x + 725 = x (x + 3) + 3 (x + 3)$

Step 2.3.1.2.2

Apply the distributive property.

$125 x + 725 = x \cdot x + x \cdot 3 + 3 (x + 3)$

Step 2.3.1.2.3

Apply the distributive property.

$125 x + 725 = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

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 $x$ by $x$ .

$125 x + 725 = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Step 2.3.1.3.1.2

Move $3$ to the left of $x$ .

$125 x + 725 = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Step 2.3.1.3.1.3

Multiply $3$ by $3$ .

$125 x + 725 = x^{2} + 3 x + 3 x + 9$

Step 2.3.1.3.2

Add $3 x$ and $3 x$ .

$125 x + 725 = x^{2} + 6 x + 9$

Step 3

Solve for $x$ .

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Step 3.1

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

$x^{2} + 6 x + 9 = 125 x + 725$

Step 3.2

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

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Step 3.2.1

Subtract $125 x$ from both sides of the equation.

$x^{2} + 6 x + 9 - 125 x = 725$

Step 3.2.2

Subtract $125 x$ from $6 x$ .

$x^{2} - 119 x + 9 = 725$

Step 3.3

Move all terms to the left side of the equation and simplify.

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Step 3.3.1

Subtract $725$ from both sides of the equation.

$x^{2} - 119 x + 9 - 725 = 0$

Step 3.3.2

Subtract $725$ from $9$ .

$x^{2} - 119 x - 716 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.5

Substitute the values $a = 1$ , $b = - 119$ , and $c = - 716$ into the quadratic formula and solve for $x$ .

$\frac{119 \pm \sqrt{{(- 119)}^{2} - 4 \cdot (1 \cdot - 716)}}{2 \cdot 1}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $- 119$ to the power of $2$ .

$x = \frac{119 \pm \sqrt{14161 - 4 \cdot 1 \cdot - 716}}{2 \cdot 1}$

Step 3.6.1.2

Multiply $- 4 \cdot 1 \cdot - 716$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{119 \pm \sqrt{14161 - 4 \cdot - 716}}{2 \cdot 1}$

Step 3.6.1.2.2

Multiply $- 4$ by $- 716$ .

$x = \frac{119 \pm \sqrt{14161 + 2864}}{2 \cdot 1}$

Step 3.6.1.3

Add $14161$ and $2864$ .

$x = \frac{119 \pm \sqrt{17025}}{2 \cdot 1}$

Step 3.6.1.4

Rewrite $17025$ as $5^{2} \cdot 681$ .

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Step 3.6.1.4.1

Factor $25$ out of $17025$ .

$x = \frac{119 \pm \sqrt{25 (681)}}{2 \cdot 1}$

Step 3.6.1.4.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{119 \pm \sqrt{5^{2} \cdot 681}}{2 \cdot 1}$

Step 3.6.1.5

Pull terms out from under the radical.

$x = \frac{119 \pm 5 \sqrt{681}}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$x = \frac{119 \pm 5 \sqrt{681}}{2}$

Step 3.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $- 119$ to the power of $2$ .

$x = \frac{119 \pm \sqrt{14161 - 4 \cdot 1 \cdot - 716}}{2 \cdot 1}$

Step 3.7.1.2

Multiply $- 4 \cdot 1 \cdot - 716$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{119 \pm \sqrt{14161 - 4 \cdot - 716}}{2 \cdot 1}$

Step 3.7.1.2.2

Multiply $- 4$ by $- 716$ .

$x = \frac{119 \pm \sqrt{14161 + 2864}}{2 \cdot 1}$

Step 3.7.1.3

Add $14161$ and $2864$ .

$x = \frac{119 \pm \sqrt{17025}}{2 \cdot 1}$

Step 3.7.1.4

Rewrite $17025$ as $5^{2} \cdot 681$ .

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Step 3.7.1.4.1

Factor $25$ out of $17025$ .

$x = \frac{119 \pm \sqrt{25 (681)}}{2 \cdot 1}$

Step 3.7.1.4.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{119 \pm \sqrt{5^{2} \cdot 681}}{2 \cdot 1}$

Step 3.7.1.5

Pull terms out from under the radical.

$x = \frac{119 \pm 5 \sqrt{681}}{2 \cdot 1}$

Step 3.7.2

Multiply $2$ by $1$ .

$x = \frac{119 \pm 5 \sqrt{681}}{2}$

Step 3.7.3

Change the $\pm$ to $+$ .

$x = \frac{119 + 5 \sqrt{681}}{2}$

Step 3.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

Raise $- 119$ to the power of $2$ .

$x = \frac{119 \pm \sqrt{14161 - 4 \cdot 1 \cdot - 716}}{2 \cdot 1}$

Step 3.8.1.2

Multiply $- 4 \cdot 1 \cdot - 716$ .

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Step 3.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{119 \pm \sqrt{14161 - 4 \cdot - 716}}{2 \cdot 1}$

Step 3.8.1.2.2

Multiply $- 4$ by $- 716$ .

$x = \frac{119 \pm \sqrt{14161 + 2864}}{2 \cdot 1}$

Step 3.8.1.3

Add $14161$ and $2864$ .

$x = \frac{119 \pm \sqrt{17025}}{2 \cdot 1}$

Step 3.8.1.4

Rewrite $17025$ as $5^{2} \cdot 681$ .

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Step 3.8.1.4.1

Factor $25$ out of $17025$ .

$x = \frac{119 \pm \sqrt{25 (681)}}{2 \cdot 1}$

Step 3.8.1.4.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{119 \pm \sqrt{5^{2} \cdot 681}}{2 \cdot 1}$

Step 3.8.1.5

Pull terms out from under the radical.

$x = \frac{119 \pm 5 \sqrt{681}}{2 \cdot 1}$

Step 3.8.2

Multiply $2$ by $1$ .

$x = \frac{119 \pm 5 \sqrt{681}}{2}$

Step 3.8.3

Change the $\pm$ to $-$ .

$x = \frac{119 - 5 \sqrt{681}}{2}$

Step 3.9

The final answer is the combination of both solutions.

$x = \frac{119 + 5 \sqrt{681}}{2}, \frac{119 - 5 \sqrt{681}}{2}$

Step 4

Exclude the solutions that do not make $5 \sqrt{5 x + 29} = x + 3$ true.

$x = \frac{119 + 5 \sqrt{681}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{119 + 5 \sqrt{681}}{2}$

Decimal Form:

$x = 124.73994175 \dots$