Precalculus Examples

$3 x^{\frac{1}{2}} + 5 x^{\frac{1}{4}} - 2 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $x^{\frac{1}{2}}$ as ${(x^{\frac{1}{4}})}^{2}$ .

$3 {(x^{\frac{1}{4}})}^{2} + 5 x^{\frac{1}{4}} - 2 = 0$

Step 1.2

Let $u = x^{\frac{1}{4}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{4}}$ .

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

Step 1.3

Factor by grouping.

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Step 1.3.1

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 1.3.1.1

Factor $5$ out of $5 u$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Apply the distributive property.

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

Step 1.3.2

Factor out the greatest common factor from each group.

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Step 1.3.2.1

Group the first two terms and the last two terms.

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

Step 1.3.2.2

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

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

Step 1.3.3

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

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

Step 1.4

Replace all occurrences of $u$ with $x^{\frac{1}{4}}$ .

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

Step 2

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

$3 x^{\frac{1}{4}} - 1 = 0$

$x^{\frac{1}{4}} + 2 = 0$

Step 3

Set $3 x^{\frac{1}{4}} - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3 x^{\frac{1}{4}} - 1$ equal to $0$ .

$3 x^{\frac{1}{4}} - 1 = 0$

Step 3.2

Solve $3 x^{\frac{1}{4}} - 1 = 0$ for $x$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

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

Step 3.2.2

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

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

Step 3.2.3

Simplify the exponent.

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Step 3.2.3.1

Simplify the left side.

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Step 3.2.3.1.1

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

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Step 3.2.3.1.1.1

Apply the product rule to $3 x^{\frac{1}{4}}$ .

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

Step 3.2.3.1.1.2

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

$81 {(x^{\frac{1}{4}})}^{4} = 1^{4}$

Step 3.2.3.1.1.3

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

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Step 3.2.3.1.1.3.1

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

$81 x^{\frac{1}{4} \cdot 4} = 1^{4}$

Step 3.2.3.1.1.3.2

Cancel the common factor of $4$ .

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Step 3.2.3.1.1.3.2.1

Cancel the common factor.

$81 x^{\frac{1}{4} \cdot 4} = 1^{4}$

Step 3.2.3.1.1.3.2.2

Rewrite the expression.

$81 x^{1} = 1^{4}$

Step 3.2.3.1.1.4

Simplify.

$81 x = 1^{4}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

One to any power is one.

$81 x = 1$

Step 3.2.4

Divide each term in $81 x = 1$ by $81$ and simplify.

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Step 3.2.4.1

Divide each term in $81 x = 1$ by $81$ .

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

Step 3.2.4.2

Simplify the left side.

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Step 3.2.4.2.1

Cancel the common factor of $81$ .

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Step 3.2.4.2.1.1

Cancel the common factor.

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

Step 3.2.4.2.1.2

Divide $x$ by $1$ .

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

Step 4

Set $x^{\frac{1}{4}} + 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{\frac{1}{4}} + 2$ equal to $0$ .

$x^{\frac{1}{4}} + 2 = 0$

Step 4.2

Solve $x^{\frac{1}{4}} + 2 = 0$ for $x$ .

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Step 4.2.1

Subtract $2$ from both sides of the equation.

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

Step 4.2.2

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 4.2.3

Simplify the exponent.

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Step 4.2.3.1

Simplify the left side.

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Step 4.2.3.1.1

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

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Step 4.2.3.1.1.1

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

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Step 4.2.3.1.1.1.1

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

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

Step 4.2.3.1.1.1.2

Cancel the common factor of $4$ .

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Step 4.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 4.2.3.1.1.2

Simplify.

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

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

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

$x = 16$

Step 5

The final solution is all the values that make $(3 x^{\frac{1}{4}} - 1) (x^{\frac{1}{4}} + 2) = 0$ true.

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

Step 6

Exclude the solutions that do not make $3 x^{\frac{1}{2}} + 5 x^{\frac{1}{4}} - 2 = 0$ true.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.01234567 \dots$