Precalculus Examples

$\sqrt{2 \sqrt{7 x + 2}} = \sqrt{4 x + 2}$

Step 1

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

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

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 \sqrt{7 x + 2})}^{1} = {\sqrt{4 x + 2}}^{2}$

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{4 x + 2}}^{2}$ .

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Step 2.3.1.1

Factor $2$ out of $4 x + 2$ .

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Step 2.3.1.1.1

Factor $2$ out of $4 x$ .

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

Step 2.3.1.1.2

Factor $2$ out of $2$ .

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

Step 2.3.1.1.3

Factor $2$ out of $2 (2 x) + 2 (1)$ .

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

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

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

Step 2.3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 2.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

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

Step 2.3.1.2.4.2

Rewrite the expression.

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

Step 2.3.1.2.5

Simplify.

$2 \sqrt{7 x + 2} = 2 (2 x + 1)$

Step 2.3.1.3

Apply the distributive property.

$2 \sqrt{7 x + 2} = 2 (2 x) + 2 \cdot 1$

Step 2.3.1.4

Multiply.

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Step 2.3.1.4.1

Multiply $2$ by $2$ .

$2 \sqrt{7 x + 2} = 4 x + 2 \cdot 1$

Step 2.3.1.4.2

Multiply $2$ by $1$ .

$2 \sqrt{7 x + 2} = 4 x + 2$

Step 3

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

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

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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Step 4.2.1.3.1

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

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

Step 4.2.1.3.2

Cancel the common factor of $2$ .

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Step 4.2.1.3.2.1

Cancel the common factor.

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

Step 4.2.1.3.2.2

Rewrite the expression.

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

Step 4.2.1.4

Simplify.

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

Step 4.2.1.5

Apply the distributive property.

$4 (7 x) + 4 \cdot 2 = {(4 x + 2)}^{2}$

Step 4.2.1.6

Multiply.

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Step 4.2.1.6.1

Multiply $7$ by $4$ .

$28 x + 4 \cdot 2 = {(4 x + 2)}^{2}$

Step 4.2.1.6.2

Multiply $4$ by $2$ .

$28 x + 8 = {(4 x + 2)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

Rewrite ${(4 x + 2)}^{2}$ as $(4 x + 2) (4 x + 2)$ .

$28 x + 8 = (4 x + 2) (4 x + 2)$

Step 4.3.1.2

Expand $(4 x + 2) (4 x + 2)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$28 x + 8 = 4 x (4 x + 2) + 2 (4 x + 2)$

Step 4.3.1.2.2

Apply the distributive property.

$28 x + 8 = 4 x (4 x) + 4 x \cdot 2 + 2 (4 x + 2)$

Step 4.3.1.2.3

Apply the distributive property.

$28 x + 8 = 4 x (4 x) + 4 x \cdot 2 + 2 (4 x) + 2 \cdot 2$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$28 x + 8 = 4 \cdot 4 x \cdot x + 4 x \cdot 2 + 2 (4 x) + 2 \cdot 2$

Step 4.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.1.3.1.2.1

Move $x$ .

$28 x + 8 = 4 \cdot 4 (x \cdot x) + 4 x \cdot 2 + 2 (4 x) + 2 \cdot 2$

Step 4.3.1.3.1.2.2

Multiply $x$ by $x$ .

$28 x + 8 = 4 \cdot 4 x^{2} + 4 x \cdot 2 + 2 (4 x) + 2 \cdot 2$

Step 4.3.1.3.1.3

Multiply $4$ by $4$ .

$28 x + 8 = 16 x^{2} + 4 x \cdot 2 + 2 (4 x) + 2 \cdot 2$

Step 4.3.1.3.1.4

Multiply $2$ by $4$ .

$28 x + 8 = 16 x^{2} + 8 x + 2 (4 x) + 2 \cdot 2$

Step 4.3.1.3.1.5

Multiply $4$ by $2$ .

$28 x + 8 = 16 x^{2} + 8 x + 8 x + 2 \cdot 2$

Step 4.3.1.3.1.6

Multiply $2$ by $2$ .

$28 x + 8 = 16 x^{2} + 8 x + 8 x + 4$

Step 4.3.1.3.2

Add $8 x$ and $8 x$ .

$28 x + 8 = 16 x^{2} + 16 x + 4$

Step 5

Solve for $x$ .

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Step 5.1

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

$16 x^{2} + 16 x + 4 = 28 x + 8$

Step 5.2

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

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Step 5.2.1

Subtract $28 x$ from both sides of the equation.

$16 x^{2} + 16 x + 4 - 28 x = 8$

Step 5.2.2

Subtract $28 x$ from $16 x$ .

$16 x^{2} - 12 x + 4 = 8$

Step 5.3

Subtract $8$ from both sides of the equation.

$16 x^{2} - 12 x + 4 - 8 = 0$

Step 5.4

Subtract $8$ from $4$ .

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

Step 5.5

Factor the left side of the equation.

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Step 5.5.1

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

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Step 5.5.1.1

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

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

Step 5.5.1.2

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

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

Step 5.5.1.3

Factor $4$ out of $- 4$ .

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

Step 5.5.1.4

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

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

Step 5.5.1.5

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

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

Step 5.5.2

Factor.

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Step 5.5.2.1

Factor by grouping.

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Step 5.5.2.1.1

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 5.5.2.1.1.1

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

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

Step 5.5.2.1.1.2

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

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

Step 5.5.2.1.1.3

Apply the distributive property.

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

Step 5.5.2.1.1.4

Multiply $x$ by $1$ .

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

Step 5.5.2.1.2

Factor out the greatest common factor from each group.

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Step 5.5.2.1.2.1

Group the first two terms and the last two terms.

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

Step 5.5.2.1.2.2

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

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

Step 5.5.2.1.3

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

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

Step 5.5.2.2

Remove unnecessary parentheses.

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

Step 5.6

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 5.7

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

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Step 5.7.1

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

$4 x + 1 = 0$

Step 5.7.2

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

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Step 5.7.2.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 5.7.2.2

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

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Step 5.7.2.2.1

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

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

Step 5.7.2.2.2

Simplify the left side.

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Step 5.7.2.2.2.1

Cancel the common factor of $4$ .

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Step 5.7.2.2.2.1.1

Cancel the common factor.

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

Step 5.7.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.7.2.2.3

Simplify the right side.

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Step 5.7.2.2.3.1

Move the negative in front of the fraction.

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

Step 5.8

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

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Step 5.8.1

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

$x - 1 = 0$

Step 5.8.2

Add $1$ to both sides of the equation.

$x = 1$

Step 5.9

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.25, 1$