Finite Math Examples

$(0, 75)$ , $(50, 175)$ , $(100, 350)$

Step 1

Use the standard form of a quadratic equation $y = a x^{2} + b x + c$ as the starting point for finding the equation through the three points.

$y = a x^{2} + b x + c$

Step 2

Create a system of equations by substituting the $x$ and $y$ values of each point into the standard formula of a quadratic equation to create the three equation system.

$75 = a {(0)}^{2} + b (0) + c, 175 = a {(50)}^{2} + b (50) + c, 350 = a {(100)}^{2} + b (100) + c$

Step 3

Solve the system of equations to find the values of $a$ , $b$ , and $c$ .

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Step 3.1

Solve for $c$ in $75 = a \cdot 0^{2} + c$ .

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Step 3.1.1

Rewrite the equation as $a \cdot 0^{2} + c = 75$ .

$a \cdot 0^{2} + c = 75$

$175 = a \cdot 50^{2} + b (50) + c$

$350 = a \cdot 100^{2} + b (100) + c$

Step 3.1.2

Simplify $a \cdot 0^{2} + c$ .

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Raising $0$ to any positive power yields $0$ .

$a \cdot 0 + c = 75$

$175 = a \cdot 50^{2} + b (50) + c$

$350 = a \cdot 100^{2} + b (100) + c$

Step 3.1.2.1.2

Multiply $a$ by $0$ .

$0 + c = 75$

$175 = a \cdot 50^{2} + b (50) + c$

$350 = a \cdot 100^{2} + b (100) + c$

$0 + c = 75$

$175 = a \cdot 50^{2} + b (50) + c$

$350 = a \cdot 100^{2} + b (100) + c$

Step 3.1.2.2

Add $0$ and $c$ .

$c = 75$

$175 = a \cdot 50^{2} + b (50) + c$

$350 = a \cdot 100^{2} + b (100) + c$

$c = 75$

$175 = a \cdot 50^{2} + b (50) + c$

$350 = a \cdot 100^{2} + b (100) + c$

$c = 75$

$175 = a \cdot 50^{2} + b (50) + c$

$350 = a \cdot 100^{2} + b (100) + c$

Step 3.2

Replace all occurrences of $c$ with $75$ in each equation.

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Step 3.2.1

Replace all occurrences of $c$ in $175 = a \cdot 50^{2} + b (50) + c$ with $75$ .

$175 = a \cdot 50^{2} + b (50) + 75$

$c = 75$

$350 = a \cdot 100^{2} + b (100) + c$

Step 3.2.2

Simplify $175 = a \cdot 50^{2} + b (50) + 75$ .

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Remove parentheses.

$175 = a \cdot 50^{2} + b (50) + 75$

$c = 75$

$350 = a \cdot 100^{2} + b (100) + c$

$175 = a \cdot 50^{2} + b (50) + 75$

$c = 75$

$350 = a \cdot 100^{2} + b (100) + c$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Simplify each term.

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Step 3.2.2.2.1.1

Raise $50$ to the power of $2$ .

$175 = a \cdot 2500 + b (50) + 75$

$c = 75$

$350 = a \cdot 100^{2} + b (100) + c$

Step 3.2.2.2.1.2

Move $2500$ to the left of $a$ .

$175 = 2500 \cdot a + b (50) + 75$

$c = 75$

$350 = a \cdot 100^{2} + b (100) + c$

Step 3.2.2.2.1.3

Move $50$ to the left of $b$ .

$175 = 2500 a + 50 b + 75$

$c = 75$

$350 = a \cdot 100^{2} + b (100) + c$

$175 = 2500 a + 50 b + 75$

$c = 75$

$350 = a \cdot 100^{2} + b (100) + c$

$175 = 2500 a + 50 b + 75$

$c = 75$

$350 = a \cdot 100^{2} + b (100) + c$

$175 = 2500 a + 50 b + 75$

$c = 75$

$350 = a \cdot 100^{2} + b (100) + c$

Step 3.2.3

Replace all occurrences of $c$ in $350 = a \cdot 100^{2} + b (100) + c$ with $75$ .

$350 = a \cdot 100^{2} + b (100) + 75$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.2.4

Simplify $350 = a \cdot 100^{2} + b (100) + 75$ .

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Step 3.2.4.1

Simplify the left side.

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Step 3.2.4.1.1

Remove parentheses.

$350 = a \cdot 100^{2} + b (100) + 75$

$175 = 2500 a + 50 b + 75$

$c = 75$

$350 = a \cdot 100^{2} + b (100) + 75$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.2.4.2

Simplify the right side.

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Step 3.2.4.2.1

Simplify each term.

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Step 3.2.4.2.1.1

Raise $100$ to the power of $2$ .

$350 = a \cdot 10000 + b (100) + 75$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.2.4.2.1.2

Move $10000$ to the left of $a$ .

$350 = 10000 \cdot a + b (100) + 75$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.2.4.2.1.3

Move $100$ to the left of $b$ .

$350 = 10000 a + 100 b + 75$

$175 = 2500 a + 50 b + 75$

$c = 75$

$350 = 10000 a + 100 b + 75$

$175 = 2500 a + 50 b + 75$

$c = 75$

$350 = 10000 a + 100 b + 75$

$175 = 2500 a + 50 b + 75$

$c = 75$

$350 = 10000 a + 100 b + 75$

$175 = 2500 a + 50 b + 75$

$c = 75$

$350 = 10000 a + 100 b + 75$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3

Solve for $a$ in $350 = 10000 a + 100 b + 75$ .

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Step 3.3.1

Rewrite the equation as $10000 a + 100 b + 75 = 350$ .

$10000 a + 100 b + 75 = 350$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $100 b$ from both sides of the equation.

$10000 a + 75 = 350 - 100 b$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.2.2

Subtract $75$ from both sides of the equation.

$10000 a = 350 - 100 b - 75$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.2.3

Subtract $75$ from $350$ .

$10000 a = - 100 b + 275$

$175 = 2500 a + 50 b + 75$

$c = 75$

$10000 a = - 100 b + 275$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3

Divide each term in $10000 a = - 100 b + 275$ by $10000$ and simplify.

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Step 3.3.3.1

Divide each term in $10000 a = - 100 b + 275$ by $10000$ .

$\frac{10000 a}{10000} = \frac{- 100 b}{10000} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $10000$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{10000 a}{10000} = \frac{- 100 b}{10000} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 100 b}{10000} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

$a = \frac{- 100 b}{10000} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

$a = \frac{- 100 b}{10000} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Simplify each term.

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Step 3.3.3.3.1.1

Cancel the common factor of $- 100$ and $10000$ .

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Step 3.3.3.3.1.1.1

Factor $100$ out of $- 100 b$ .

$a = \frac{100 (- b)}{10000} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3.3.1.1.2

Cancel the common factors.

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Step 3.3.3.3.1.1.2.1

Factor $100$ out of $10000$ .

$a = \frac{100 (- b)}{100 (100)} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3.3.1.1.2.2

Cancel the common factor.

$a = \frac{100 (- b)}{100 \cdot 100} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{100} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

$a = \frac{- b}{100} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

$a = \frac{- b}{100} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{100} + \frac{275}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3.3.1.3

Cancel the common factor of $275$ and $10000$ .

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Step 3.3.3.3.1.3.1

Factor $25$ out of $275$ .

$a = - \frac{b}{100} + \frac{25 (11)}{10000}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3.3.1.3.2

Cancel the common factors.

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Step 3.3.3.3.1.3.2.1

Factor $25$ out of $10000$ .

$a = - \frac{b}{100} + \frac{25 \cdot 11}{25 \cdot 400}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3.3.1.3.2.2

Cancel the common factor.

$a = - \frac{b}{100} + \frac{25 \cdot 11}{25 \cdot 400}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.3.3.3.1.3.2.3

Rewrite the expression.

$a = - \frac{b}{100} + \frac{11}{400}$

$175 = 2500 a + 50 b + 75$

$c = 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$175 = 2500 a + 50 b + 75$

$c = 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$175 = 2500 a + 50 b + 75$

$c = 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$175 = 2500 a + 50 b + 75$

$c = 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$175 = 2500 a + 50 b + 75$

$c = 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$175 = 2500 a + 50 b + 75$

$c = 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$175 = 2500 a + 50 b + 75$

$c = 75$

Step 3.4

Replace all occurrences of $a$ with $- \frac{b}{100} + \frac{11}{400}$ in each equation.

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Step 3.4.1

Replace all occurrences of $a$ in $175 = 2500 a + 50 b + 75$ with $- \frac{b}{100} + \frac{11}{400}$ .

$175 = 2500 (- \frac{b}{100} + \frac{11}{400}) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $2500 (- \frac{b}{100} + \frac{11}{400}) + 50 b + 75$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Apply the distributive property.

$175 = 2500 (- \frac{b}{100}) + 2500 (\frac{11}{400}) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.1.2

Cancel the common factor of $100$ .

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Step 3.4.2.1.1.2.1

Move the leading negative in $- \frac{b}{100}$ into the numerator.

$175 = 2500 (\frac{- b}{100}) + 2500 (\frac{11}{400}) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.1.2.2

Factor $100$ out of $2500$ .

$175 = 100 (25) (\frac{- b}{100}) + 2500 (\frac{11}{400}) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.1.2.3

Cancel the common factor.

$175 = 100 \cdot (25 (\frac{- b}{100})) + 2500 (\frac{11}{400}) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.1.2.4

Rewrite the expression.

$175 = 25 (- b) + 2500 (\frac{11}{400}) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$175 = 25 (- b) + 2500 (\frac{11}{400}) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.1.3

Multiply $- 1$ by $25$ .

$175 = - 25 b + 2500 (\frac{11}{400}) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.1.4

Cancel the common factor of $100$ .

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Step 3.4.2.1.1.4.1

Factor $100$ out of $2500$ .

$175 = - 25 b + 100 (25) (\frac{11}{400}) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.1.4.2

Factor $100$ out of $400$ .

$175 = - 25 b + 100 \cdot (25 (\frac{11}{100 \cdot 4})) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.1.4.3

Cancel the common factor.

$175 = - 25 b + 100 \cdot (25 (\frac{11}{100 \cdot 4})) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.1.4.4

Rewrite the expression.

$175 = - 25 b + 25 (\frac{11}{4}) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$175 = - 25 b + 25 (\frac{11}{4}) + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.1.5

Combine $25$ and $\frac{11}{4}$ .

$175 = - 25 b + \frac{25 \cdot 11}{4} + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.1.6

Multiply $25$ by $11$ .

$175 = - 25 b + \frac{275}{4} + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$175 = - 25 b + \frac{275}{4} + 50 b + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.2

Add $- 25 b$ and $50 b$ .

$175 = 25 b + \frac{275}{4} + 75$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.3

To write $75$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$175 = 25 b + \frac{275}{4} + 75 \cdot \frac{4}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.4

Combine $75$ and $\frac{4}{4}$ .

$175 = 25 b + \frac{275}{4} + \frac{75 \cdot 4}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.5

Combine the numerators over the common denominator.

$175 = 25 b + \frac{275 + 75 \cdot 4}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.6

Simplify the numerator.

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Step 3.4.2.1.6.1

Multiply $75$ by $4$ .

$175 = 25 b + \frac{275 + 300}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.4.2.1.6.2

Add $275$ and $300$ .

$175 = 25 b + \frac{575}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$175 = 25 b + \frac{575}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$175 = 25 b + \frac{575}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$175 = 25 b + \frac{575}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$175 = 25 b + \frac{575}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5

Solve for $b$ in $175 = 25 b + \frac{575}{4}$ .

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Step 3.5.1

Rewrite the equation as $25 b + \frac{575}{4} = 175$ .

$25 b + \frac{575}{4} = 175$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.2

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

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Step 3.5.2.1

Subtract $\frac{575}{4}$ from both sides of the equation.

$25 b = 175 - \frac{575}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.2.2

To write $175$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$25 b = 175 \cdot \frac{4}{4} - \frac{575}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.2.3

Combine $175$ and $\frac{4}{4}$ .

$25 b = \frac{175 \cdot 4}{4} - \frac{575}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.2.4

Combine the numerators over the common denominator.

$25 b = \frac{175 \cdot 4 - 575}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.2.5

Simplify the numerator.

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Step 3.5.2.5.1

Multiply $175$ by $4$ .

$25 b = \frac{700 - 575}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.2.5.2

Subtract $575$ from $700$ .

$25 b = \frac{125}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$25 b = \frac{125}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$25 b = \frac{125}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.3

Divide each term in $25 b = \frac{125}{4}$ by $25$ and simplify.

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Step 3.5.3.1

Divide each term in $25 b = \frac{125}{4}$ by $25$ .

$\frac{25 b}{25} = \frac{\frac{125}{4}}{25}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $25$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{25 b}{25} = \frac{\frac{125}{4}}{25}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{\frac{125}{4}}{25}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$b = \frac{\frac{125}{4}}{25}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$b = \frac{\frac{125}{4}}{25}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$b = \frac{125}{4} \cdot \frac{1}{25}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.3.3.2

Cancel the common factor of $25$ .

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Step 3.5.3.3.2.1

Factor $25$ out of $125$ .

$b = \frac{25 (5)}{4} \cdot \frac{1}{25}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.3.3.2.2

Cancel the common factor.

$b = \frac{25 \cdot 5}{4} \cdot \frac{1}{25}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.5.3.3.2.3

Rewrite the expression.

$b = \frac{5}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$b = \frac{5}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$b = \frac{5}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$b = \frac{5}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

$b = \frac{5}{4}$

$a = - \frac{b}{100} + \frac{11}{400}$

$c = 75$

Step 3.6

Replace all occurrences of $b$ with $\frac{5}{4}$ in each equation.

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Step 3.6.1

Replace all occurrences of $b$ in $a = - \frac{b}{100} + \frac{11}{400}$ with $\frac{5}{4}$ .

$a = - \frac{\frac{5}{4}}{100} + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $- \frac{\frac{5}{4}}{100} + \frac{11}{400}$ .

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Step 3.6.2.1.1

Simplify each term.

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Step 3.6.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$a = - (\frac{5}{4} \cdot \frac{1}{100}) + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.6.2.1.1.2.1

Factor $5$ out of $100$ .

$a = - (\frac{5}{4} \cdot \frac{1}{5 (20)}) + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.1.2.2

Cancel the common factor.

$a = - (\frac{5}{4} \cdot \frac{1}{5 \cdot 20}) + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.1.2.3

Rewrite the expression.

$a = - (\frac{1}{4} \cdot \frac{1}{20}) + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

$a = - (\frac{1}{4} \cdot \frac{1}{20}) + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.1.3

Multiply $\frac{1}{4}$ by $\frac{1}{20}$ .

$a = - \frac{1}{4 \cdot 20} + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.1.4

Multiply $4$ by $20$ .

$a = - \frac{1}{80} + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

$a = - \frac{1}{80} + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.2

To write $- \frac{1}{80}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$a = - \frac{1}{80} \cdot \frac{5}{5} + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.3

Write each expression with a common denominator of $400$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.6.2.1.3.1

Multiply $\frac{1}{80}$ by $\frac{5}{5}$ .

$a = - \frac{5}{80 \cdot 5} + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.3.2

Multiply $80$ by $5$ .

$a = - \frac{5}{400} + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

$a = - \frac{5}{400} + \frac{11}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.4

Simplify the expression.

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Step 3.6.2.1.4.1

Combine the numerators over the common denominator.

$a = \frac{- 5 + 11}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.4.2

Add $- 5$ and $11$ .

$a = \frac{6}{400}$

$b = \frac{5}{4}$

$c = 75$

$a = \frac{6}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.5

Cancel the common factor of $6$ and $400$ .

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Step 3.6.2.1.5.1

Factor $2$ out of $6$ .

$a = \frac{2 (3)}{400}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.5.2

Cancel the common factors.

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Step 3.6.2.1.5.2.1

Factor $2$ out of $400$ .

$a = \frac{2 \cdot 3}{2 \cdot 200}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.5.2.2

Cancel the common factor.

$a = \frac{2 \cdot 3}{2 \cdot 200}$

$b = \frac{5}{4}$

$c = 75$

Step 3.6.2.1.5.2.3

Rewrite the expression.

$a = \frac{3}{200}$

$b = \frac{5}{4}$

$c = 75$

$a = \frac{3}{200}$

$b = \frac{5}{4}$

$c = 75$

$a = \frac{3}{200}$

$b = \frac{5}{4}$

$c = 75$

$a = \frac{3}{200}$

$b = \frac{5}{4}$

$c = 75$

$a = \frac{3}{200}$

$b = \frac{5}{4}$

$c = 75$

$a = \frac{3}{200}$

$b = \frac{5}{4}$

$c = 75$

Step 3.7

List all of the solutions.

$a = \frac{3}{200}, b = \frac{5}{4}, c = 75$

Step 4

Substitute the actual values of $a$ , $b$ , and $c$ into the formula for a quadratic equation to find the resulting equation.

$y = \frac{3 x^{2}}{200} + \frac{5 x}{4} + 75$

Step 5