Calculus Examples

$\frac{d y}{d t} = (1 + 4 t) \sqrt{y}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\sqrt{y}}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = \frac{1}{\sqrt{y}} ((1 + 4 t) \sqrt{y})$

Step 1.2

Cancel the common factor of $\sqrt{y}$ .

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Step 1.2.1

Factor $\sqrt{y}$ out of $(1 + 4 t) \sqrt{y}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = \frac{1}{\sqrt{y}} (\sqrt{y} (1 + 4 t))$

Step 1.2.2

Cancel the common factor.

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = \frac{1}{\sqrt{y}} (\sqrt{y} (1 + 4 t))$

Step 1.2.3

Rewrite the expression.

$\frac{1}{\sqrt{y}} \frac{d y}{d t} = 1 + 4 t$

Step 1.3

Rewrite the equation.

$\frac{1}{\sqrt{y}} d y = (1 + 4 t) d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{y}} d y = \int 1 + 4 t d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\int \frac{1}{y^{\frac{1}{2}}} d y = \int 1 + 4 t d t$

Step 2.2.1.2

Move $y^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{\frac{1}{2}})}^{- 1} d y = \int 1 + 4 t d t$

Step 2.2.1.3

Multiply the exponents in ${(y^{\frac{1}{2}})}^{- 1}$ .

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Step 2.2.1.3.1

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

$\int y^{\frac{1}{2} \cdot - 1} d y = \int 1 + 4 t d t$

Step 2.2.1.3.2

Combine $\frac{1}{2}$ and $- 1$ .

$\int y^{\frac{- 1}{2}} d y = \int 1 + 4 t d t$

Step 2.2.1.3.3

Move the negative in front of the fraction.

$\int y^{- \frac{1}{2}} d y = \int 1 + 4 t d t$

Step 2.2.2

By the Power Rule, the integral of $y^{- \frac{1}{2}}$ with respect to $y$ is $2 y^{\frac{1}{2}}$ .

$2 y^{\frac{1}{2}} + C_{1} = \int 1 + 4 t d t$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$2 y^{\frac{1}{2}} + C_{1} = \int d t + \int 4 t d t$

Step 2.3.2

Apply the constant rule.

$2 y^{\frac{1}{2}} + C_{1} = t + C_{2} + \int 4 t d t$

Step 2.3.3

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

$2 y^{\frac{1}{2}} + C_{1} = t + C_{2} + 4 \int t d t$

Step 2.3.4

By the Power Rule, the integral of $t$ with respect to $t$ is $\frac{1}{2} t^{2}$ .

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Simplify.

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

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

Combine $4$ and $\frac{1}{2}$ .

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

Step 2.3.5.2.2

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

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Step 2.3.5.2.2.1

Factor $2$ out of $4$ .

$2 y^{\frac{1}{2}} + C_{1} = t + \frac{2 \cdot 2}{2} t^{2} + C_{4}$

Step 2.3.5.2.2.2

Cancel the common factors.

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Step 2.3.5.2.2.2.1

Factor $2$ out of $2$ .

$2 y^{\frac{1}{2}} + C_{1} = t + \frac{2 \cdot 2}{2 (1)} t^{2} + C_{4}$

Step 2.3.5.2.2.2.2

Cancel the common factor.

$2 y^{\frac{1}{2}} + C_{1} = t + \frac{2 \cdot 2}{2 \cdot 1} t^{2} + C_{4}$

Step 2.3.5.2.2.2.3

Rewrite the expression.

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

Step 2.3.5.2.2.2.4

Divide $2$ by $1$ .

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

Step 2.3.6

Reorder terms.

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

Step 2.4

Group the constant of integration on the right side as $K$ .

$2 y^{\frac{1}{2}} = 2 t^{2} + t + K$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 y^{\frac{1}{2}} = 2 t^{2} + t + K$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 y^{\frac{1}{2}} = 2 t^{2} + t + K$ by $2$ .

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{2 t^{2}}{2} + \frac{t}{2} + \frac{K}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{2 t^{2}}{2} + \frac{t}{2} + \frac{K}{2}$

Step 3.1.2.2

Divide $y^{\frac{1}{2}}$ by $1$ .

$y^{\frac{1}{2}} = \frac{2 t^{2}}{2} + \frac{t}{2} + \frac{K}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Cancel the common factor of $2$ .

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Step 3.1.3.1.1

Cancel the common factor.

$y^{\frac{1}{2}} = \frac{2 t^{2}}{2} + \frac{t}{2} + \frac{K}{2}$

Step 3.1.3.1.2

Divide $t^{2}$ by $1$ .

$y^{\frac{1}{2}} = t^{2} + \frac{t}{2} + \frac{K}{2}$

Step 3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(y^{\frac{1}{2}})}^{2} = {(t^{2} + \frac{t}{2} + \frac{K}{2})}^{2}$

Step 3.3

Simplify the exponent.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = {(t^{2} + \frac{t}{2} + \frac{K}{2})}^{2}$

Step 3.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(t^{2} + \frac{t}{2} + \frac{K}{2})}^{2}$

Step 3.3.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(t^{2} + \frac{t}{2} + \frac{K}{2})}^{2}$

Step 3.3.1.1.2

Simplify.

$y = {(t^{2} + \frac{t}{2} + \frac{K}{2})}^{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify ${(t^{2} + \frac{t}{2} + \frac{K}{2})}^{2}$ .

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Step 3.3.2.1.1

Rewrite ${(t^{2} + \frac{t}{2} + \frac{K}{2})}^{2}$ as $(t^{2} + \frac{t}{2} + \frac{K}{2}) (t^{2} + \frac{t}{2} + \frac{K}{2})$ .

$y = (t^{2} + \frac{t}{2} + \frac{K}{2}) (t^{2} + \frac{t}{2} + \frac{K}{2})$

Step 3.3.2.1.2

Expand $(t^{2} + \frac{t}{2} + \frac{K}{2}) (t^{2} + \frac{t}{2} + \frac{K}{2})$ by multiplying each term in the first expression by each term in the second expression.

$y = t^{2} t^{2} + t^{2} \frac{t}{2} + t^{2} \frac{K}{2} + \frac{t}{2} t^{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3

Simplify terms.

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Step 3.3.2.1.3.1

Simplify each term.

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Step 3.3.2.1.3.1.1

Multiply $t^{2}$ by $t^{2}$ by adding the exponents.

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Step 3.3.2.1.3.1.1.1

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

$y = t^{2 + 2} + t^{2} \frac{t}{2} + t^{2} \frac{K}{2} + \frac{t}{2} t^{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.1.2

Add $2$ and $2$ .

$y = t^{4} + t^{2} \frac{t}{2} + t^{2} \frac{K}{2} + \frac{t}{2} t^{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2

Multiply $t^{2} \frac{t}{2}$ .

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Step 3.3.2.1.3.1.2.1

Combine $t^{2}$ and $\frac{t}{2}$ .

$y = t^{4} + \frac{t^{2} t}{2} + t^{2} \frac{K}{2} + \frac{t}{2} t^{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2.2

Multiply $t^{2}$ by $t$ by adding the exponents.

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Step 3.3.2.1.3.1.2.2.1

Multiply $t^{2}$ by $t$ .

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Step 3.3.2.1.3.1.2.2.1.1

Raise $t$ to the power of $1$ .

$y = t^{4} + \frac{t^{2} t^{1}}{2} + t^{2} \frac{K}{2} + \frac{t}{2} t^{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2.2.1.2

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

$y = t^{4} + \frac{t^{2 + 1}}{2} + t^{2} \frac{K}{2} + \frac{t}{2} t^{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2.2.2

Add $2$ and $1$ .

$y = t^{4} + \frac{t^{3}}{2} + t^{2} \frac{K}{2} + \frac{t}{2} t^{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.3

Combine $t^{2}$ and $\frac{K}{2}$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t}{2} t^{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.4

Multiply $\frac{t}{2} t^{2}$ .

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Step 3.3.2.1.3.1.4.1

Combine $\frac{t}{2}$ and $t^{2}$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t \cdot t^{2}}{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.4.2

Multiply $t$ by $t^{2}$ by adding the exponents.

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Step 3.3.2.1.3.1.4.2.1

Multiply $t$ by $t^{2}$ .

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Step 3.3.2.1.3.1.4.2.1.1

Raise $t$ to the power of $1$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{1} t^{2}}{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.4.2.1.2

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

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{1 + 2}}{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.4.2.2

Add $1$ and $2$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t}{2} \cdot \frac{t}{2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.5

Multiply $\frac{t}{2} \cdot \frac{t}{2}$ .

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Step 3.3.2.1.3.1.5.1

Multiply $\frac{t}{2}$ by $\frac{t}{2}$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t \cdot t}{2 \cdot 2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.5.2

Raise $t$ to the power of $1$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{1} t}{2 \cdot 2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.5.3

Raise $t$ to the power of $1$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{1} t^{1}}{2 \cdot 2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.5.4

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

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{1 + 1}}{2 \cdot 2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.5.5

Add $1$ and $1$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{2 \cdot 2} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.5.6

Multiply $2$ by $2$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t}{2} \cdot \frac{K}{2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.6

Multiply $\frac{t}{2} \cdot \frac{K}{2}$ .

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Step 3.3.2.1.3.1.6.1

Multiply $\frac{t}{2}$ by $\frac{K}{2}$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t K}{2 \cdot 2} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.6.2

Multiply $2$ by $2$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t K}{4} + \frac{K}{2} t^{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.7

Combine $\frac{K}{2}$ and $t^{2}$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t K}{4} + \frac{K t^{2}}{2} + \frac{K}{2} \cdot \frac{t}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.8

Multiply $\frac{K}{2} \cdot \frac{t}{2}$ .

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Step 3.3.2.1.3.1.8.1

Multiply $\frac{K}{2}$ by $\frac{t}{2}$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t K}{4} + \frac{K t^{2}}{2} + \frac{K t}{2 \cdot 2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.8.2

Multiply $2$ by $2$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t K}{4} + \frac{K t^{2}}{2} + \frac{K t}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.9

Multiply $\frac{K}{2} \cdot \frac{K}{2}$ .

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Step 3.3.2.1.3.1.9.1

Multiply $\frac{K}{2}$ by $\frac{K}{2}$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t K}{4} + \frac{K t^{2}}{2} + \frac{K t}{4} + \frac{K \cdot K}{2 \cdot 2}$

Step 3.3.2.1.3.1.9.2

Raise $K$ to the power of $1$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t K}{4} + \frac{K t^{2}}{2} + \frac{K t}{4} + \frac{K^{1} K}{2 \cdot 2}$

Step 3.3.2.1.3.1.9.3

Raise $K$ to the power of $1$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t K}{4} + \frac{K t^{2}}{2} + \frac{K t}{4} + \frac{K^{1} K^{1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.9.4

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

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t K}{4} + \frac{K t^{2}}{2} + \frac{K t}{4} + \frac{K^{1 + 1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.9.5

Add $1$ and $1$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t K}{4} + \frac{K t^{2}}{2} + \frac{K t}{4} + \frac{K^{2}}{2 \cdot 2}$

Step 3.3.2.1.3.1.9.6

Multiply $2$ by $2$ .

$y = t^{4} + \frac{t^{3}}{2} + \frac{t^{2} K}{2} + \frac{t^{3}}{2} + \frac{t^{2}}{4} + \frac{t K}{4} + \frac{K t^{2}}{2} + \frac{K t}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2

Simplify terms.

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Step 3.3.2.1.3.2.1

Combine the numerators over the common denominator.

$y = t^{4} + \frac{t^{3} + t^{2} K + t^{3} + K t^{2}}{2} + \frac{t^{2} + t K + K t + K^{2}}{4}$

Step 3.3.2.1.3.2.2

Add $t^{3}$ and $t^{3}$ .

$y = t^{4} + \frac{2 t^{3} + t^{2} K + K t^{2}}{2} + \frac{t^{2} + t K + K t + K^{2}}{4}$

Step 3.3.2.1.4

Add $t^{2} K$ and $K t^{2}$ .

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Step 3.3.2.1.4.1

Reorder $t^{2}$ and $K$ .

$y = t^{4} + \frac{2 t^{3} + K t^{2} + K t^{2}}{2} + \frac{t^{2} + t K + K t + K^{2}}{4}$

Step 3.3.2.1.4.2

Add $K t^{2}$ and $K t^{2}$ .

$y = t^{4} + \frac{2 t^{3} + 2 K t^{2}}{2} + \frac{t^{2} + t K + K t + K^{2}}{4}$

Step 3.3.2.1.5

Add $t K$ and $K t$ .

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Step 3.3.2.1.5.1

Reorder $t$ and $K$ .

$y = t^{4} + \frac{2 t^{3} + 2 K t^{2}}{2} + \frac{t^{2} + K t + K t + K^{2}}{4}$

Step 3.3.2.1.5.2

Add $K t$ and $K t$ .

$y = t^{4} + \frac{2 t^{3} + 2 K t^{2}}{2} + \frac{t^{2} + 2 K t + K^{2}}{4}$

Step 3.3.2.1.6

Simplify each term.

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Step 3.3.2.1.6.1

Factor $2 t^{2}$ out of $2 t^{3} + 2 K t^{2}$ .

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Step 3.3.2.1.6.1.1

Factor $2 t^{2}$ out of $2 t^{3}$ .

$y = t^{4} + \frac{2 t^{2} (t) + 2 K t^{2}}{2} + \frac{t^{2} + 2 K t + K^{2}}{4}$

Step 3.3.2.1.6.1.2

Factor $2 t^{2}$ out of $2 K t^{2}$ .

$y = t^{4} + \frac{2 t^{2} (t) + 2 t^{2} K}{2} + \frac{t^{2} + 2 K t + K^{2}}{4}$

Step 3.3.2.1.6.1.3

Factor $2 t^{2}$ out of $2 t^{2} (t) + 2 t^{2} K$ .

$y = t^{4} + \frac{2 t^{2} (t + K)}{2} + \frac{t^{2} + 2 K t + K^{2}}{4}$

Step 3.3.2.1.6.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.6.2.1

Cancel the common factor.

$y = t^{4} + \frac{2 t^{2} (t + K)}{2} + \frac{t^{2} + 2 K t + K^{2}}{4}$

Step 3.3.2.1.6.2.2

Divide $t^{2} (t + K)$ by $1$ .

$y = t^{4} + t^{2} (t + K) + \frac{t^{2} + 2 K t + K^{2}}{4}$

Step 3.3.2.1.6.3

Apply the distributive property.

$y = t^{4} + t^{2} t + t^{2} K + \frac{t^{2} + 2 K t + K^{2}}{4}$

Step 3.3.2.1.6.4

Multiply $t^{2}$ by $t$ by adding the exponents.

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Step 3.3.2.1.6.4.1

Multiply $t^{2}$ by $t$ .

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Step 3.3.2.1.6.4.1.1

Raise $t$ to the power of $1$ .

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

Step 3.3.2.1.6.4.1.2

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

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

Step 3.3.2.1.6.4.2

Add $2$ and $1$ .

$y = t^{4} + t^{3} + t^{2} K + \frac{t^{2} + 2 K t + K^{2}}{4}$

Step 3.3.2.1.6.5

Factor using the perfect square rule.

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Step 3.3.2.1.6.5.1

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

$2 K t = 2 \cdot t \cdot K$

Step 3.3.2.1.6.5.2

Rewrite the polynomial.

$y = t^{4} + t^{3} + t^{2} K + \frac{t^{2} + 2 \cdot t \cdot K + K^{2}}{4}$

Step 3.3.2.1.6.5.3

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

$y = t^{4} + t^{3} + t^{2} K + \frac{{(t + K)}^{2}}{4}$

Step 3.4

Simplify $t^{4} + t^{3} + t^{2} K + \frac{{(t + K)}^{2}}{4}$ .

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Step 3.4.1

Reorder $t^{2}$ and $K$ .

$y = t^{4} + t^{3} + K t^{2} + \frac{{(t + K)}^{2}}{4}$

Step 3.4.2

Reorder $t$ and $K$ .

$y = t^{4} + t^{3} + K t^{2} + \frac{{(K + t)}^{2}}{4}$

Step 3.4.3

Move $t^{3}$ .

$y = t^{4} + K t^{2} + \frac{{(K + t)}^{2}}{4} + t^{3}$

Step 3.4.4

Move $K t^{2}$ .

$y = t^{4} + \frac{{(K + t)}^{2}}{4} + K t^{2} + t^{3}$

Step 3.4.5

Reorder $t^{4}$ and $\frac{{(K + t)}^{2}}{4}$ .

$y = \frac{{(K + t)}^{2}}{4} + t^{4} + K t^{2} + t^{3}$