Precalculus Examples

$2 x = \sqrt{- 10 x - 4}$

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.

$\sqrt{- 10 x - 4} = 2 x$

Step 2

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

${\sqrt{- 10 x - 4}}^{2} = {(2 x)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 10 x - 4}$ as ${(- 10 x - 4)}^{\frac{1}{2}}$ .

${({(- 10 x - 4)}^{\frac{1}{2}})}^{2} = {(2 x)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(- 10 x - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(- 10 x - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(- 10 x - 4)}^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(- 10 x - 4)}^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(- 10 x - 4)}^{1} = {(2 x)}^{2}$

Step 3.2.1.2

Simplify.

$- 10 x - 4 = {(2 x)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Apply the product rule to $2 x$ .

$- 10 x - 4 = 2^{2} x^{2}$

Step 3.3.1.2

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

$- 10 x - 4 = 4 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

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

$- 10 x - 4 - 4 x^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Factor $- 2$ out of $- 10 x - 4 - 4 x^{2}$ .

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move $- 4$ .

$- 10 x - 4 x^{2} - 4 = 0$

Step 4.2.1.1.2

Reorder $- 10 x$ and $- 4 x^{2}$ .

$- 4 x^{2} - 10 x - 4 = 0$

Step 4.2.1.2

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

$- 2 (2 x^{2}) - 10 x - 4 = 0$

Step 4.2.1.3

Factor $- 2$ out of $- 10 x$ .

$- 2 (2 x^{2}) - 2 (5 x) - 4 = 0$

Step 4.2.1.4

Factor $- 2$ out of $- 4$ .

$- 2 (2 x^{2}) - 2 (5 x) - 2 \cdot 2 = 0$

Step 4.2.1.5

Factor $- 2$ out of $- 2 (2 x^{2}) - 2 (5 x)$ .

$- 2 (2 x^{2} + 5 x) - 2 \cdot 2 = 0$

Step 4.2.1.6

Factor $- 2$ out of $- 2 (2 x^{2} + 5 x) - 2 (2)$ .

$- 2 (2 x^{2} + 5 x + 2) = 0$

Step 4.2.2

Factor.

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Step 4.2.2.1

Factor by grouping.

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Step 4.2.2.1.1

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 $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = 5$ .

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Step 4.2.2.1.1.1

Factor $5$ out of $5 x$ .

$- 2 (2 x^{2} + 5 (x) + 2) = 0$

Step 4.2.2.1.1.2

Rewrite $5$ as $1$ plus $4$

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

Step 4.2.2.1.1.3

Apply the distributive property.

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

Step 4.2.2.1.1.4

Multiply $x$ by $1$ .

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

Step 4.2.2.1.2

Factor out the greatest common factor from each group.

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Step 4.2.2.1.2.1

Group the first two terms and the last two terms.

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

Step 4.2.2.1.2.2

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

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

Step 4.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

Step 4.2.2.2

Remove unnecessary parentheses.

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

Step 4.3

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

$2 x + 1 = 0$

$x + 2 = 0$

Step 4.4

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 4.4.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 4.4.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 4.4.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 4.4.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

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

Step 4.4.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.4.2.2.2.1.1

Cancel the common factor.

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

Step 4.4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.4.2.2.3

Simplify the right side.

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Step 4.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 4.5

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

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Step 4.5.1

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

$x + 2 = 0$

Step 4.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.6

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

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

Step 5

Exclude the solutions that do not make $2 x = \sqrt{- 10 x - 4}$ true.

$No solution$