Algebra Examples

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

Step 1

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

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

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

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

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.

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

Step 2.2.1.3.2.2

Rewrite the expression.

$4 {(3 x + 1)}^{1} = {(4 x)}^{2}$

Step 2.2.1.4

Simplify.

$4 (3 x + 1) = {(4 x)}^{2}$

Step 2.2.1.5

Apply the distributive property.

$4 (3 x) + 4 \cdot 1 = {(4 x)}^{2}$

Step 2.2.1.6

Multiply.

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Step 2.2.1.6.1

Multiply $3$ by $4$ .

$12 x + 4 \cdot 1 = {(4 x)}^{2}$

Step 2.2.1.6.2

Multiply $4$ by $1$ .

$12 x + 4 = {(4 x)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(4 x)}^{2}$ .

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Step 2.3.1.1

Apply the product rule to $4 x$ .

$12 x + 4 = 4^{2} x^{2}$

Step 2.3.1.2

Raise $4$ to the power of $2$ .

$12 x + 4 = 16 x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $16 x^{2}$ from both sides of the equation.

$12 x + 4 - 16 x^{2} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $- 4$ out of $12 x + 4 - 16 x^{2}$ .

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Step 3.2.1.1

Reorder the expression.

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Step 3.2.1.1.1

Move $4$ .

$12 x - 16 x^{2} + 4 = 0$

Step 3.2.1.1.2

Reorder $12 x$ and $- 16 x^{2}$ .

$- 16 x^{2} + 12 x + 4 = 0$

Step 3.2.1.2

Factor $- 4$ out of $- 16 x^{2}$ .

$- 4 (4 x^{2}) + 12 x + 4 = 0$

Step 3.2.1.3

Factor $- 4$ out of $12 x$ .

$- 4 (4 x^{2}) - 4 (- 3 x) + 4 = 0$

Step 3.2.1.4

Factor $- 4$ out of $4$ .

$- 4 (4 x^{2}) - 4 (- 3 x) - 4 \cdot - 1 = 0$

Step 3.2.1.5

Factor $- 4$ out of $- 4 (4 x^{2}) - 4 (- 3 x)$ .

$- 4 (4 x^{2} - 3 x) - 4 \cdot - 1 = 0$

Step 3.2.1.6

Factor $- 4$ out of $- 4 (4 x^{2} - 3 x) - 4 (- 1)$ .

$- 4 (4 x^{2} - 3 x - 1) = 0$

Step 3.2.2

Factor.

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Step 3.2.2.1

Factor by grouping.

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Step 3.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 1 = - 4$ and whose sum is $b = - 3$ .

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Step 3.2.2.1.1.1

Factor $- 3$ out of $- 3 x$ .

$- 4 (4 x^{2} - 3 x - 1) = 0$

Step 3.2.2.1.1.2

Rewrite $- 3$ as $1$ plus $- 4$

$- 4 (4 x^{2} + (1 - 4) x - 1) = 0$

Step 3.2.2.1.1.3

Apply the distributive property.

$- 4 (4 x^{2} + 1 x - 4 x - 1) = 0$

Step 3.2.2.1.1.4

Multiply $x$ by $1$ .

$- 4 (4 x^{2} + x - 4 x - 1) = 0$

Step 3.2.2.1.2

Factor out the greatest common factor from each group.

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Step 3.2.2.1.2.1

Group the first two terms and the last two terms.

$- 4 ((4 x^{2} + x) - 4 x - 1) = 0$

Step 3.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- 4 (x (4 x + 1) - (4 x + 1)) = 0$

Step 3.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $4 x + 1$ .

$- 4 ((4 x + 1) (x - 1)) = 0$

Step 3.2.2.2

Remove unnecessary parentheses.

$- 4 (4 x + 1) (x - 1) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 x + 1 = 0$

$x - 1 = 0$

Step 3.4

Set $4 x + 1$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Step 3.4.2

Solve $4 x + 1 = 0$ for $x$ .

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Step 3.4.2.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 3.4.2.2

Divide each term in $4 x = - 1$ by $4$ and simplify.

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Step 3.4.2.2.1

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 3.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Step 3.4.2.2.3

Simplify the right side.

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Step 3.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Step 3.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 3.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.6

The final solution is all the values that make $- 4 (4 x + 1) (x - 1) = 0$ true.

$x = - \frac{1}{4}, 1$

Step 4

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

$x = 1$