Algebra Examples

$2 x = \sqrt{2 + 7 x}$

Step 1

Subtract $\sqrt{2 + 7 x}$ from both sides of the equation.

$2 x - \sqrt{2 + 7 x} = 0$

Step 2

Subtract $2 x$ from both sides of the equation.

$- \sqrt{2 + 7 x} = - 2 x$

Step 3

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

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

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 $\sqrt{2 + 7 x}$ as ${(2 + 7 x)}^{\frac{1}{2}}$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(- {(2 + 7 x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Apply the product rule to $- {(2 + 7 x)}^{\frac{1}{2}}$ .

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

Step 4.2.1.2

Raise $- 1$ to the power of $2$ .

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

Step 4.2.1.3

Multiply ${({(2 + 7 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

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

Step 4.2.1.4

Multiply the exponents in ${({(2 + 7 x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.4.1

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

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

Step 4.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.2.1.4.2.1

Cancel the common factor.

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

Step 4.2.1.4.2.2

Rewrite the expression.

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

Step 4.2.1.5

Simplify.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

Apply the product rule to $- 2 x$ .

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

Step 4.3.1.2

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

$2 + 7 x = 4 x^{2}$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $4 x^{2}$ from both sides of the equation.

$2 + 7 x - 4 x^{2} = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

Factor $- 1$ out of $2 + 7 x - 4 x^{2}$ .

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Step 5.2.1.1

Reorder the expression.

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Step 5.2.1.1.1

Move $2$ .

$7 x - 4 x^{2} + 2 = 0$

Step 5.2.1.1.2

Reorder $7 x$ and $- 4 x^{2}$ .

$- 4 x^{2} + 7 x + 2 = 0$

Step 5.2.1.2

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

$- (4 x^{2}) + 7 x + 2 = 0$

Step 5.2.1.3

Factor $- 1$ out of $7 x$ .

$- (4 x^{2}) - (- 7 x) + 2 = 0$

Step 5.2.1.4

Rewrite $2$ as $- 1 (- 2)$ .

$- (4 x^{2}) - (- 7 x) - 1 \cdot - 2 = 0$

Step 5.2.1.5

Factor $- 1$ out of $- (4 x^{2}) - (- 7 x)$ .

$- (4 x^{2} - 7 x) - 1 \cdot - 2 = 0$

Step 5.2.1.6

Factor $- 1$ out of $- (4 x^{2} - 7 x) - 1 (- 2)$ .

$- (4 x^{2} - 7 x - 2) = 0$

Step 5.2.2

Factor.

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Step 5.2.2.1

Factor by grouping.

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Step 5.2.2.1.1

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 $a \cdot c = 4 \cdot - 2 = - 8$ and whose sum is $b = - 7$ .

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Step 5.2.2.1.1.1

Factor $- 7$ out of $- 7 x$ .

$- (4 x^{2} - 7 x - 2) = 0$

Step 5.2.2.1.1.2

Rewrite $- 7$ as $1$ plus $- 8$

$- (4 x^{2} + (1 - 8) x - 2) = 0$

Step 5.2.2.1.1.3

Apply the distributive property.

$- (4 x^{2} + 1 x - 8 x - 2) = 0$

Step 5.2.2.1.1.4

Multiply $x$ by $1$ .

$- (4 x^{2} + x - 8 x - 2) = 0$

Step 5.2.2.1.2

Factor out the greatest common factor from each group.

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Step 5.2.2.1.2.1

Group the first two terms and the last two terms.

$- ((4 x^{2} + x) - 8 x - 2) = 0$

Step 5.2.2.1.2.2

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

$- (x (4 x + 1) - 2 (4 x + 1)) = 0$

Step 5.2.2.1.3

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

$- ((4 x + 1) (x - 2)) = 0$

Step 5.2.2.2

Remove unnecessary parentheses.

$- (4 x + 1) (x - 2) = 0$

Step 5.3

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

$4 x + 1 = 0$

$x - 2 = 0$

Step 5.4

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

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Step 5.4.1

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

$4 x + 1 = 0$

Step 5.4.2

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

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Step 5.4.2.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 5.4.2.2

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

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Step 5.4.2.2.1

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

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

Step 5.4.2.2.2

Simplify the left side.

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Step 5.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 5.4.2.2.2.1.1

Cancel the common factor.

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

Step 5.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.4.2.2.3

Simplify the right side.

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Step 5.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 5.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 5.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5.6

The final solution is all the values that make $- (4 x + 1) (x - 2) = 0$ true.

$x = - \frac{1}{4}, 2$

Step 6

Exclude the solutions that do not make $2 x - \sqrt{2 + 7 x} = 0$ true.

$x = 2$