Calculus Examples

$\frac{d y}{d x} = 6 \sqrt{y}$ , $y (1) = 16$

Step 1

Separate the variables.

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Step 1.1

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{1}{\sqrt{y}} (6 \sqrt{y})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{1}{\sqrt{y}} \sqrt{y}$

Step 1.2.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 (\frac{1}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}}) \sqrt{y}$

Step 1.2.3

Combine and simplify the denominator.

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Step 1.2.3.1

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{\sqrt{y}}{\sqrt{y} \sqrt{y}} \sqrt{y}$

Step 1.2.3.2

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{\sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}} \sqrt{y}$

Step 1.2.3.3

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{\sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}} \sqrt{y}$

Step 1.2.3.4

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{\sqrt{y}}{{\sqrt{y}}^{1 + 1}} \sqrt{y}$

Step 1.2.3.5

Add $1$ and $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{\sqrt{y}}{{\sqrt{y}}^{2}} \sqrt{y}$

Step 1.2.3.6

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 1.2.3.6.1

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{\sqrt{y}}{{(y^{\frac{1}{2}})}^{2}} \sqrt{y}$

Step 1.2.3.6.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{\sqrt{y}}{y^{\frac{1}{2} \cdot 2}} \sqrt{y}$

Step 1.2.3.6.3

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{\sqrt{y}}{y^{\frac{2}{2}}} \sqrt{y}$

Step 1.2.3.6.4

Cancel the common factor of $2$ .

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Step 1.2.3.6.4.1

Cancel the common factor.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{\sqrt{y}}{y^{\frac{2}{2}}} \sqrt{y}$

Step 1.2.3.6.4.2

Rewrite the expression.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{\sqrt{y}}{y^{1}} \sqrt{y}$

Step 1.2.3.6.5

Simplify.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6 \frac{\sqrt{y}}{y} \sqrt{y}$

Step 1.2.4

Combine $6$ and $\frac{\sqrt{y}}{y}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 \sqrt{y}}{y} \sqrt{y}$

Step 1.2.5

Multiply $\frac{6 \sqrt{y}}{y} \sqrt{y}$ .

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Step 1.2.5.1

Combine $\frac{6 \sqrt{y}}{y}$ and $\sqrt{y}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 \sqrt{y} \sqrt{y}}{y}$

Step 1.2.5.2

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 ({\sqrt{y}}^{1} \sqrt{y})}{y}$

Step 1.2.5.3

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 ({\sqrt{y}}^{1} {\sqrt{y}}^{1})}{y}$

Step 1.2.5.4

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 {\sqrt{y}}^{1 + 1}}{y}$

Step 1.2.5.5

Add $1$ and $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 {\sqrt{y}}^{2}}{y}$

Step 1.2.6

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 1.2.6.1

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 {(y^{\frac{1}{2}})}^{2}}{y}$

Step 1.2.6.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 y^{\frac{1}{2} \cdot 2}}{y}$

Step 1.2.6.3

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 y^{\frac{2}{2}}}{y}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 y^{\frac{2}{2}}}{y}$

Step 1.2.6.4.2

Rewrite the expression.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 y^{1}}{y}$

Step 1.2.6.5

Simplify.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 y}{y}$

Step 1.2.7

Cancel the common factor of $y$ .

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Step 1.2.7.1

Cancel the common factor.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{6 y}{y}$

Step 1.2.7.2

Divide $6$ by $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 6$

Step 1.3

Rewrite the equation.

$\frac{1}{\sqrt{y}} d y = 6 d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{y}} d y = \int 6 d x$

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

$\int \frac{1}{y^{\frac{1}{2}}} d y = \int 6 d x$

Step 2.2.1.2

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

$\int {(y^{\frac{1}{2}})}^{- 1} d y = \int 6 d x$

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

$\int y^{\frac{1}{2} \cdot - 1} d y = \int 6 d x$

Step 2.2.1.3.2

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

$\int y^{\frac{- 1}{2}} d y = \int 6 d x$

Step 2.2.1.3.3

Move the negative in front of the fraction.

$\int y^{- \frac{1}{2}} d y = \int 6 d x$

Step 2.2.2

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

$2 y^{\frac{1}{2}} + C_{1} = \int 6 d x$

Step 2.3

Apply the constant rule.

$2 y^{\frac{1}{2}} + C_{1} = 6 x + C_{2}$

Step 2.4

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

$2 y^{\frac{1}{2}} = 6 x + K$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 y^{\frac{1}{2}} = 6 x + K$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 y^{\frac{1}{2}} = 6 x + K$ by $2$ .

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{6 x}{2} + \frac{K}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

$\frac{2 y^{\frac{1}{2}}}{2} = \frac{6 x}{2} + \frac{K}{2}$

Step 3.1.2.2

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

$y^{\frac{1}{2}} = \frac{6 x}{2} + \frac{K}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Cancel the common factor of $6$ and $2$ .

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Step 3.1.3.1.1

Factor $2$ out of $6 x$ .

$y^{\frac{1}{2}} = \frac{2 (3 x)}{2} + \frac{K}{2}$

Step 3.1.3.1.2

Cancel the common factors.

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Step 3.1.3.1.2.1

Factor $2$ out of $2$ .

$y^{\frac{1}{2}} = \frac{2 (3 x)}{2 (1)} + \frac{K}{2}$

Step 3.1.3.1.2.2

Cancel the common factor.

$y^{\frac{1}{2}} = \frac{2 (3 x)}{2 \cdot 1} + \frac{K}{2}$

Step 3.1.3.1.2.3

Rewrite the expression.

$y^{\frac{1}{2}} = \frac{3 x}{1} + \frac{K}{2}$

Step 3.1.3.1.2.4

Divide $3 x$ by $1$ .

$y^{\frac{1}{2}} = 3 x + \frac{K}{2}$

Step 3.2

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

${(y^{\frac{1}{2}})}^{2} = {(3 x + \frac{K}{2})}^{2}$

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

$y^{\frac{1}{2} \cdot 2} = {(3 x + \frac{K}{2})}^{2}$

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.

$y^{\frac{1}{2} \cdot 2} = {(3 x + \frac{K}{2})}^{2}$

Step 3.3.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(3 x + \frac{K}{2})}^{2}$

Step 3.3.1.1.2

Simplify.

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Rewrite ${(3 x + \frac{K}{2})}^{2}$ as $(3 x + \frac{K}{2}) (3 x + \frac{K}{2})$ .

$y = (3 x + \frac{K}{2}) (3 x + \frac{K}{2})$

Step 3.3.2.1.2

Expand $(3 x + \frac{K}{2}) (3 x + \frac{K}{2})$ using the FOIL Method.

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Step 3.3.2.1.2.1

Apply the distributive property.

$y = 3 x (3 x + \frac{K}{2}) + \frac{K}{2} (3 x + \frac{K}{2})$

Step 3.3.2.1.2.2

Apply the distributive property.

$y = 3 x (3 x) + 3 x \frac{K}{2} + \frac{K}{2} (3 x + \frac{K}{2})$

Step 3.3.2.1.2.3

Apply the distributive property.

$y = 3 x (3 x) + 3 x \frac{K}{2} + \frac{K}{2} (3 x) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3

Simplify and combine like 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

Rewrite using the commutative property of multiplication.

$y = 3 \cdot 3 x \cdot x + 3 x \frac{K}{2} + \frac{K}{2} (3 x) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.2.1.3.1.2.1

Move $x$ .

$y = 3 \cdot 3 (x \cdot x) + 3 x \frac{K}{2} + \frac{K}{2} (3 x) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2.2

Multiply $x$ by $x$ .

$y = 3 \cdot 3 x^{2} + 3 x \frac{K}{2} + \frac{K}{2} (3 x) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.3

Multiply $3$ by $3$ .

$y = 9 x^{2} + 3 x \frac{K}{2} + \frac{K}{2} (3 x) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.4

Multiply $3 x \frac{K}{2}$ .

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Step 3.3.2.1.3.1.4.1

Combine $3$ and $\frac{K}{2}$ .

$y = 9 x^{2} + x \frac{3 K}{2} + \frac{K}{2} (3 x) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.4.2

Combine $x$ and $\frac{3 K}{2}$ .

$y = 9 x^{2} + \frac{x (3 K)}{2} + \frac{K}{2} (3 x) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.5

Move $3$ to the left of $x$ .

$y = 9 x^{2} + \frac{3 \cdot x K}{2} + \frac{K}{2} (3 x) + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.6

Rewrite using the commutative property of multiplication.

$y = 9 x^{2} + \frac{3 x K}{2} + 3 \frac{K}{2} x + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.7

Combine $3$ and $\frac{K}{2}$ .

$y = 9 x^{2} + \frac{3 x K}{2} + \frac{3 K}{2} x + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.8

Combine $\frac{3 K}{2}$ and $x$ .

$y = 9 x^{2} + \frac{3 x K}{2} + \frac{3 K x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

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 $\frac{K}{2}$ .

$y = 9 x^{2} + \frac{3 x K}{2} + \frac{3 K x}{2} + \frac{K \cdot K}{2 \cdot 2}$

Step 3.3.2.1.3.1.9.2

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

$y = 9 x^{2} + \frac{3 x K}{2} + \frac{3 K x}{2} + \frac{K^{1} K}{2 \cdot 2}$

Step 3.3.2.1.3.1.9.3

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

$y = 9 x^{2} + \frac{3 x K}{2} + \frac{3 K x}{2} + \frac{K^{1} K^{1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.9.4

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

$y = 9 x^{2} + \frac{3 x K}{2} + \frac{3 K x}{2} + \frac{K^{1 + 1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.9.5

Add $1$ and $1$ .

$y = 9 x^{2} + \frac{3 x K}{2} + \frac{3 K x}{2} + \frac{K^{2}}{2 \cdot 2}$

Step 3.3.2.1.3.1.9.6

Multiply $2$ by $2$ .

$y = 9 x^{2} + \frac{3 x K}{2} + \frac{3 K x}{2} + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2

Add $\frac{3 x K}{2}$ and $\frac{3 K x}{2}$ .

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Step 3.3.2.1.3.2.1

Move $x$ .

$y = 9 x^{2} + \frac{3 K x}{2} + \frac{3 K x}{2} + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2.2

Add $\frac{3 K x}{2}$ and $\frac{3 K x}{2}$ .

$y = 9 x^{2} + 2 \frac{3 K x}{2} + \frac{K^{2}}{4}$

Step 3.3.2.1.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.4.1

Cancel the common factor.

$y = 9 x^{2} + 2 \frac{3 K x}{2} + \frac{K^{2}}{4}$

Step 3.3.2.1.4.2

Rewrite the expression.

$y = 9 x^{2} + 3 K x + \frac{K^{2}}{4}$

Step 3.4

Simplify $9 x^{2} + 3 K x + \frac{K^{2}}{4}$ .

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Step 3.4.1

Move $9 x^{2}$ .

$y = 3 K x + \frac{K^{2}}{4} + 9 x^{2}$

Step 3.4.2

Reorder $3 K x$ and $\frac{K^{2}}{4}$ .

$y = \frac{K^{2}}{4} + 3 K x + 9 x^{2}$

Step 4

Simplify the constant of integration.

$y = K + K x + 9 x^{2}$

Step 5

Use the initial condition to find the value of $K$ by substituting $1$ for $x$ and $16$ for $y$ in $y = K + K x + 9 x^{2}$ .

$16 = K + K \cdot 1 + 9 \cdot 1^{2}$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $K + K \cdot 1 + 9 \cdot 1^{2} = 16$ .

$K + K \cdot 1 + 9 \cdot 1^{2} = 16$

Step 6.2

Simplify $K + K \cdot 1 + 9 \cdot 1^{2}$ .

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $K$ by $1$ .

$K + K + 9 \cdot 1^{2} = 16$

Step 6.2.1.2

One to any power is one.

$K + K + 9 \cdot 1 = 16$

Step 6.2.1.3

Multiply $9$ by $1$ .

$K + K + 9 = 16$

Step 6.2.2

Add $K$ and $K$ .

$2 K + 9 = 16$

Step 6.3

Move all terms not containing $K$ to the right side of the equation.

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Step 6.3.1

Subtract $9$ from both sides of the equation.

$2 K = 16 - 9$

Step 6.3.2

Subtract $9$ from $16$ .

$2 K = 7$

Step 6.4

Divide each term in $2 K = 7$ by $2$ and simplify.

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Step 6.4.1

Divide each term in $2 K = 7$ by $2$ .

$\frac{2 K}{2} = \frac{7}{2}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $2$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{2 K}{2} = \frac{7}{2}$

Step 6.4.2.1.2

Divide $K$ by $1$ .

$K = \frac{7}{2}$

Step 7

Substitute $\frac{7}{2}$ for $K$ in $y = K + K x + 9 x^{2}$ and simplify.

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Step 7.1

Substitute $\frac{7}{2}$ for $K$ .

$y = \frac{7}{2} + K x + 9 x^{2}$