Precalculus Examples

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

Step 1

Factor the left side of the equation.

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Step 1.1

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

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

Step 1.2

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

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

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 $3$ .

$1, 2$

Step 1.3.2

Write the factored form using these integers.

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

Step 1.4

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

$(t^{\frac{1}{4}} + 1) (t^{\frac{1}{4}} + 2) = 0$

Step 2

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

$t^{\frac{1}{4}} + 1 = 0$

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

Step 3

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

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Step 3.1

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

$t^{\frac{1}{4}} + 1 = 0$

Step 3.2

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

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$t^{\frac{1}{4}} = - 1$

Step 3.2.2

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

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

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

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Step 3.2.3.1.1.1

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

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Step 3.2.3.1.1.1.1

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

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

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.

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

Step 3.2.3.1.1.1.2.2

Rewrite the expression.

$t^{1} = {(- 1)}^{4}$

Step 3.2.3.1.1.2

Simplify.

$t = {(- 1)}^{4}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

Raise $- 1$ to the power of $4$ .

$t = 1$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Subtract $2$ from both sides of the equation.

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

Step 4.2.2

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

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

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

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Step 4.2.3.1.1.1

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

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Step 4.2.3.1.1.1.1

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

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

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.

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

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 4.2.3.1.1.2

Simplify.

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

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 $4$ .

$t = 16$

Step 5

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

$t = 1, 16$

Step 6

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

$No solution$