Precalculus Examples

$3 x^{\frac{1}{2}} + 5 x^{\frac{1}{4}} = 2$

Step 1

Find a common factor $x^{\frac{1}{4}}$ that is present in each term.

$3 {(x^{\frac{1}{4}})}^{2} + 5 x^{\frac{1}{4}} - 2$

Step 2

Substitute $u$ for $x^{\frac{1}{4}}$ .

$3 {(u)}^{2} + 5 (u) - 2 = 0$

Step 3

Solve for $u$ .

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Step 3.1

Multiply $5$ by $u$ .

$3 u^{2} + 5 u - 2 = 0$

Step 3.2

Factor by grouping.

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Step 3.2.1

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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 5$ .

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Step 3.2.1.1

Factor $5$ out of $5 u$ .

$3 u^{2} + 5 (u) - 2 = 0$

Step 3.2.1.2

Rewrite $5$ as $- 1$ plus $6$

$3 u^{2} + (- 1 + 6) u - 2 = 0$

Step 3.2.1.3

Apply the distributive property.

$3 u^{2} - 1 u + 6 u - 2 = 0$

Step 3.2.2

Factor out the greatest common factor from each group.

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Step 3.2.2.1

Group the first two terms and the last two terms.

$(3 u^{2} - 1 u) + 6 u - 2 = 0$

Step 3.2.2.2

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

$u (3 u - 1) + 2 (3 u - 1) = 0$

Step 3.2.3

Factor the polynomial by factoring out the greatest common factor, $3 u - 1$ .

$(3 u - 1) (u + 2) = 0$

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 $0$ .

$3 u - 1 = 0$

$u + 2 = 0$

Step 3.4

Set $3 u - 1$ equal to $0$ and solve for $u$ .

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Step 3.4.1

Set $3 u - 1$ equal to $0$ .

$3 u - 1 = 0$

Step 3.4.2

Solve $3 u - 1 = 0$ for $u$ .

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Step 3.4.2.1

Add $1$ to both sides of the equation.

$3 u = 1$

Step 3.4.2.2

Divide each term in $3 u = 1$ by $3$ and simplify.

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Step 3.4.2.2.1

Divide each term in $3 u = 1$ by $3$ .

$\frac{3 u}{3} = \frac{1}{3}$

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 $3$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 u}{3} = \frac{1}{3}$

Step 3.4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{3}$

Step 3.5

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

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Step 3.5.1

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

$u + 2 = 0$

Step 3.5.2

Subtract $2$ from both sides of the equation.

$u = - 2$

Step 3.6

The final solution is all the values that make $(3 u - 1) (u + 2) = 0$ true.

$u = \frac{1}{3}, - 2$

Step 4

Substitute $x$ for $u$ .

$x^{\frac{1}{4}} = \frac{1}{3}, - 2$

Step 5

Solve for $x^{\frac{1}{4}} = \frac{1}{3}$ for $x$ .

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Step 5.1

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

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

Step 5.2

Simplify the exponent.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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Step 5.2.1.1.1.1

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

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

Step 5.2.1.1.1.2

Cancel the common factor of $4$ .

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Step 5.2.1.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.1.2.2

Rewrite the expression.

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

Step 5.2.1.1.2

Simplify.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify ${(\frac{1}{3})}^{4}$ .

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Step 5.2.2.1.1

Apply the product rule to $\frac{1}{3}$ .

$x = \frac{1^{4}}{3^{4}}$

Step 5.2.2.1.2

One to any power is one.

$x = \frac{1}{3^{4}}$

Step 5.2.2.1.3

Raise $3$ to the power of $4$ .

$x = \frac{1}{81}$

Step 6

Solve for $x^{\frac{1}{4}} = - 2$ for $x$ .

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Step 6.1

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

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

Step 6.2

Simplify the exponent.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

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

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Step 6.2.1.1.1

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

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Step 6.2.1.1.1.1

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

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

Step 6.2.1.1.1.2

Cancel the common factor of $4$ .

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Step 6.2.1.1.1.2.1

Cancel the common factor.

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

Step 6.2.1.1.1.2.2

Rewrite the expression.

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

Step 6.2.1.1.2

Simplify.

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

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

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

$x = 16$

Step 7

List all of the solutions.

$x = \frac{1}{81}, 16$

Step 8

Exclude the solutions that do not make $3 x^{\frac{1}{2}} + 5 x^{\frac{1}{4}} = 2$ true.

$x = \frac{1}{81}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{81}$

Decimal Form:

$x = 0.01234567 \dots$