Algebra Examples

$t^{\frac{1}{2}} - 10 t^{\frac{1}{4}} + 25 = 0$

Step 1

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

${(t^{\frac{1}{4}})}^{2} - 10 t^{\frac{1}{4}} + 25 = 0$

Step 2

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

$u^{2} - 10 u + 25 = 0$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $25$ as $5^{2}$ .

$u^{2} - 10 u + 5^{2} = 0$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 u = 2 \cdot u \cdot 5$

Step 3.3

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 5 + 5^{2} = 0$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 5$ .

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

Step 4

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

${(t^{\frac{1}{4}} - 5)}^{2} = 0$

Step 5

Simplify ${(t^{\frac{1}{4}} - 5)}^{2}$ .

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Step 5.1

Rewrite ${(t^{\frac{1}{4}} - 5)}^{2}$ as $(t^{\frac{1}{4}} - 5) (t^{\frac{1}{4}} - 5)$ .

$(t^{\frac{1}{4}} - 5) (t^{\frac{1}{4}} - 5) = 0$

Step 5.2

Expand $(t^{\frac{1}{4}} - 5) (t^{\frac{1}{4}} - 5)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

$t^{\frac{1}{4}} (t^{\frac{1}{4}} - 5) - 5 (t^{\frac{1}{4}} - 5) = 0$

Step 5.2.2

Apply the distributive property.

$t^{\frac{1}{4}} t^{\frac{1}{4}} + t^{\frac{1}{4}} \cdot - 5 - 5 (t^{\frac{1}{4}} - 5) = 0$

Step 5.2.3

Apply the distributive property.

$t^{\frac{1}{4}} t^{\frac{1}{4}} + t^{\frac{1}{4}} \cdot - 5 - 5 t^{\frac{1}{4}} - 5 \cdot - 5 = 0$

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $t^{\frac{1}{4}}$ by $t^{\frac{1}{4}}$ by adding the exponents.

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Step 5.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$t^{\frac{1}{4} + \frac{1}{4}} + t^{\frac{1}{4}} \cdot - 5 - 5 t^{\frac{1}{4}} - 5 \cdot - 5 = 0$

Step 5.3.1.1.2

Combine the numerators over the common denominator.

$t^{\frac{1 + 1}{4}} + t^{\frac{1}{4}} \cdot - 5 - 5 t^{\frac{1}{4}} - 5 \cdot - 5 = 0$

Step 5.3.1.1.3

Add $1$ and $1$ .

$t^{\frac{2}{4}} + t^{\frac{1}{4}} \cdot - 5 - 5 t^{\frac{1}{4}} - 5 \cdot - 5 = 0$

Step 5.3.1.1.4

Cancel the common factor of $2$ and $4$ .

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Step 5.3.1.1.4.1

Factor $2$ out of $2$ .

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

Step 5.3.1.1.4.2

Cancel the common factors.

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Step 5.3.1.1.4.2.1

Factor $2$ out of $4$ .

$t^{\frac{2 \cdot 1}{2 \cdot 2}} + t^{\frac{1}{4}} \cdot - 5 - 5 t^{\frac{1}{4}} - 5 \cdot - 5 = 0$

Step 5.3.1.1.4.2.2

Cancel the common factor.

$t^{\frac{2 \cdot 1}{2 \cdot 2}} + t^{\frac{1}{4}} \cdot - 5 - 5 t^{\frac{1}{4}} - 5 \cdot - 5 = 0$

Step 5.3.1.1.4.2.3

Rewrite the expression.

$t^{\frac{1}{2}} + t^{\frac{1}{4}} \cdot - 5 - 5 t^{\frac{1}{4}} - 5 \cdot - 5 = 0$

Step 5.3.1.2

Move $- 5$ to the left of $t^{\frac{1}{4}}$ .

$t^{\frac{1}{2}} - 5 \cdot t^{\frac{1}{4}} - 5 t^{\frac{1}{4}} - 5 \cdot - 5 = 0$

Step 5.3.1.3

Multiply $- 5$ by $- 5$ .

$t^{\frac{1}{2}} - 5 t^{\frac{1}{4}} - 5 t^{\frac{1}{4}} + 25 = 0$

Step 5.3.2

Subtract $5 t^{\frac{1}{4}}$ from $- 5 t^{\frac{1}{4}}$ .

$t^{\frac{1}{2}} - 10 t^{\frac{1}{4}} + 25 = 0$

Step 6

Graph each side of the equation. The solution is the x-value of the point of intersection.

No solution