Precalculus Examples

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

Step 1

Factor the left side of the equation.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Factor $u^{2} - 3 u + 2$ using the AC method.

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Step 1.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 1.3.2

Write the factored form using these integers.

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

Step 1.4

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

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

Step 2

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

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Add $2$ to both sides of the equation.

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

Step 3.2.2

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

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Step 3.2.3.1.1.1

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

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Step 3.2.3.1.1.1.1

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

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

Step 3.2.3.1.1.1.2

Cancel the common factor of $4$ .

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Step 3.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 3.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 2^{4}$

Step 3.2.3.1.1.2

Simplify.

$x = 2^{4}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

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

$x = 16$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Add $1$ to both sides of the equation.

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

Step 4.2.2

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

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.

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

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 4.2.3.1.1.2

Simplify.

$x = 1^{4}$

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

One to any power is one.

$x = 1$

Step 5

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

$x = 16, 1$