Find the Equation of the Parabola (1,8) , (2,3) , (4,-31)

$(1, 8)$ , $(2, 3)$ , $(4, - 31)$

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.

$8 = a {(1)}^{2} + b (1) + c, 3 = a {(2)}^{2} + b (2) + c, - 31 = a {(4)}^{2} + b (4) + c$

Step 3

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

Solve for $a$ in $8 = a + b + c$ .

Rewrite the equation as $a + b + c = 8$ .

$a + b + c = 8$

$3 = a \cdot 2^{2} + b (2) + c$

$- 31 = a \cdot 4^{2} + b (4) + c$

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

Subtract $b$ from both sides of the equation.

$a + c = 8 - b$

$3 = a \cdot 2^{2} + b (2) + c$

$- 31 = a \cdot 4^{2} + b (4) + c$

Subtract $c$ from both sides of the equation.

$a = 8 - b - c$

$3 = a \cdot 2^{2} + b (2) + c$

$- 31 = a \cdot 4^{2} + b (4) + c$

$a = 8 - b - c$

$3 = a \cdot 2^{2} + b (2) + c$

$- 31 = a \cdot 4^{2} + b (4) + c$

$a = 8 - b - c$

$3 = a \cdot 2^{2} + b (2) + c$

$- 31 = a \cdot 4^{2} + b (4) + c$

Replace all occurrences of $a$ with $8 - b - c$ in each equation.

Replace all occurrences of $a$ in $3 = a \cdot 2^{2} + b (2) + c$ with $8 - b - c$ .

$3 = (8 - b - c) \cdot 2^{2} + b (2) + c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

Simplify the right side.

Simplify $(8 - b - c) \cdot 2^{2} + b (2) + c$ .

Simplify each term.

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

$3 = (8 - b - c) \cdot 4 + b (2) + c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

Apply the distributive property.

$3 = 8 \cdot 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

Simplify.

Multiply $8$ by $4$ .

$3 = 32 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

Multiply $4$ by $- 1$ .

$3 = 32 - 4 b - c \cdot 4 + b (2) + c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

Multiply $4$ by $- 1$ .

$3 = 32 - 4 b - 4 c + b (2) + c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

$3 = 32 - 4 b - 4 c + b (2) + c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

Move $2$ to the left of $b$ .

$3 = 32 - 4 b - 4 c + 2 b + c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

$3 = 32 - 4 b - 4 c + 2 b + c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

Simplify by adding terms.

Add $- 4 b$ and $2 b$ .

$3 = 32 - 2 b - 4 c + c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

Add $- 4 c$ and $c$ .

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$- 31 = a \cdot 4^{2} + b (4) + c$

Replace all occurrences of $a$ in $- 31 = a \cdot 4^{2} + b (4) + c$ with $8 - b - c$ .

$- 31 = (8 - b - c) \cdot 4^{2} + b (4) + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Simplify the right side.

Simplify $(8 - b - c) \cdot 4^{2} + b (4) + c$ .

Simplify each term.

Raise $4$ to the power of $2$ .

$- 31 = (8 - b - c) \cdot 16 + b (4) + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Apply the distributive property.

$- 31 = 8 \cdot 16 - b \cdot 16 - c \cdot 16 + b (4) + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Simplify.

Multiply $8$ by $16$ .

$- 31 = 128 - b \cdot 16 - c \cdot 16 + b (4) + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Multiply $16$ by $- 1$ .

$- 31 = 128 - 16 b - c \cdot 16 + b (4) + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Multiply $16$ by $- 1$ .

$- 31 = 128 - 16 b - 16 c + b (4) + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$- 31 = 128 - 16 b - 16 c + b (4) + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Move $4$ to the left of $b$ .

$- 31 = 128 - 16 b - 16 c + 4 b + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$- 31 = 128 - 16 b - 16 c + 4 b + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Simplify by adding terms.

Add $- 16 b$ and $4 b$ .

$- 31 = 128 - 12 b - 16 c + c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Add $- 16 c$ and $c$ .

$- 31 = 128 - 12 b - 15 c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$- 31 = 128 - 12 b - 15 c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$- 31 = 128 - 12 b - 15 c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$- 31 = 128 - 12 b - 15 c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$- 31 = 128 - 12 b - 15 c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Solve for $b$ in $- 31 = 128 - 12 b - 15 c$ .

Rewrite the equation as $128 - 12 b - 15 c = - 31$ .

$128 - 12 b - 15 c = - 31$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

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

Subtract $128$ from both sides of the equation.

$- 12 b - 15 c = - 31 - 128$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Add $15 c$ to both sides of the equation.

$- 12 b = - 31 - 128 + 15 c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Subtract $128$ from $- 31$ .

$- 12 b = - 159 + 15 c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$- 12 b = - 159 + 15 c$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Divide each term in $- 12 b = - 159 + 15 c$ by $- 12$ and simplify.

Divide each term in $- 12 b = - 159 + 15 c$ by $- 12$ .

$\frac{- 12 b}{- 12} = \frac{- 159}{- 12} + \frac{15 c}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 b}{- 12} = \frac{- 159}{- 12} + \frac{15 c}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Divide $b$ by $1$ .

$b = \frac{- 159}{- 12} + \frac{15 c}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$b = \frac{- 159}{- 12} + \frac{15 c}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$b = \frac{- 159}{- 12} + \frac{15 c}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 159$ and $- 12$ .

Factor $- 3$ out of $- 159$ .

$b = \frac{- 3 \cdot 53}{- 12} + \frac{15 c}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Cancel the common factors.

Factor $- 3$ out of $- 12$ .

$b = \frac{- 3 \cdot 53}{- 3 \cdot 4} + \frac{15 c}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Cancel the common factor.

$b = \frac{- 3 \cdot 53}{- 3 \cdot 4} + \frac{15 c}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Rewrite the expression.

$b = \frac{53}{4} + \frac{15 c}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$b = \frac{53}{4} + \frac{15 c}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$b = \frac{53}{4} + \frac{15 c}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Cancel the common factor of $15$ and $- 12$ .

Factor $3$ out of $15 c$ .

$b = \frac{53}{4} + \frac{3 (5 c)}{- 12}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Cancel the common factors.

Factor $3$ out of $- 12$ .

$b = \frac{53}{4} + \frac{3 (5 c)}{3 (- 4)}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Cancel the common factor.

$b = \frac{53}{4} + \frac{3 (5 c)}{3 \cdot - 4}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Rewrite the expression.

$b = \frac{53}{4} + \frac{5 c}{- 4}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$b = \frac{53}{4} + \frac{5 c}{- 4}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$b = \frac{53}{4} + \frac{5 c}{- 4}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Move the negative in front of the fraction.

$b = \frac{53}{4} - \frac{5 c}{4}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$3 = 32 - 2 b - 3 c$

$a = 8 - b - c$

Replace all occurrences of $b$ with $\frac{53}{4} - \frac{5 c}{4}$ in each equation.

Replace all occurrences of $b$ in $3 = 32 - 2 b - 3 c$ with $\frac{53}{4} - \frac{5 c}{4}$ .

$3 = 32 - 2 (\frac{53}{4} - \frac{5 c}{4}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Simplify the right side.

Simplify $32 - 2 (\frac{53}{4} - \frac{5 c}{4}) - 3 c$ .

Simplify each term.

Apply the distributive property.

$3 = 32 - 2 (\frac{53}{4}) - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Cancel the common factor of $2$ .

Factor $2$ out of $- 2$ .

$3 = 32 + 2 (- 1) (\frac{53}{4}) - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Factor $2$ out of $4$ .

$3 = 32 + 2 \cdot (- 1 \frac{53}{2 \cdot 2}) - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Cancel the common factor.

$3 = 32 + 2 \cdot (- 1 \frac{53}{2 \cdot 2}) - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Rewrite the expression.

$3 = 32 - 1 (\frac{53}{2}) - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

$3 = 32 - 1 (\frac{53}{2}) - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{5 c}{4}$ into the numerator.

$3 = 32 - 1 (\frac{53}{2}) - 2 \frac{- 5 c}{4} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Factor $2$ out of $- 2$ .

$3 = 32 - 1 (\frac{53}{2}) + 2 (- 1) (\frac{- 5 c}{4}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Factor $2$ out of $4$ .

$3 = 32 - 1 (\frac{53}{2}) + 2 \cdot (- 1 \frac{- 5 c}{2 \cdot 2}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Cancel the common factor.

$3 = 32 - 1 (\frac{53}{2}) + 2 \cdot (- 1 \frac{- 5 c}{2 \cdot 2}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Rewrite the expression.

$3 = 32 - 1 (\frac{53}{2}) - 1 \frac{- 5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

$3 = 32 - 1 (\frac{53}{2}) - 1 \frac{- 5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Simplify each term.

Rewrite $- 1 (\frac{53}{2})$ as $- (\frac{53}{2})$ .

$3 = 32 - (\frac{53}{2}) - 1 \frac{- 5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Move the negative in front of the fraction.

$3 = 32 - \frac{53}{2} - 1 (- \frac{(5) c}{2}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Multiply $- 1 (- \frac{5 c}{2})$ .

Multiply $- 1$ by $- 1$ .

$3 = 32 - \frac{53}{2} + 1 (\frac{5 c}{2}) - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Multiply $\frac{5 c}{2}$ by $1$ .

$3 = 32 - \frac{53}{2} + \frac{5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

$3 = 32 - \frac{53}{2} + \frac{5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

$3 = 32 - \frac{53}{2} + \frac{5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

$3 = 32 - \frac{53}{2} + \frac{5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

To write $32$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 = 32 \cdot \frac{2}{2} - \frac{53}{2} + \frac{5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Combine $32$ and $\frac{2}{2}$ .

$3 = \frac{32 \cdot 2}{2} - \frac{53}{2} + \frac{5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Combine the numerators over the common denominator.

$3 = \frac{32 \cdot 2 - 53}{2} + \frac{5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Simplify the numerator.

Multiply $32$ by $2$ .

$3 = \frac{64 - 53}{2} + \frac{5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Subtract $53$ from $64$ .

$3 = \frac{11}{2} + \frac{5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

$3 = \frac{11}{2} + \frac{5 c}{2} - 3 c$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

To write $- 3 c$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 = \frac{11}{2} + \frac{5 c}{2} - 3 c \cdot \frac{2}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Combine $- 3 c$ and $\frac{2}{2}$ .

$3 = \frac{11}{2} + \frac{5 c}{2} + \frac{- 3 c \cdot 2}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Combine the numerators over the common denominator.

$3 = \frac{11}{2} + \frac{5 c - 3 c \cdot 2}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Combine the numerators over the common denominator.

$3 = \frac{11 + 5 c - 3 c \cdot 2}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Multiply $2$ by $- 3$ .

$3 = \frac{11 + 5 c - 6 c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Subtract $6 c$ from $5 c$ .

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - b - c$

Replace all occurrences of $b$ in $a = 8 - b - c$ with $\frac{53}{4} - \frac{5 c}{4}$ .

$a = 8 - (\frac{53}{4} - \frac{5 c}{4}) - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Simplify the right side.

Simplify $8 - (\frac{53}{4} - \frac{5 c}{4}) - c$ .

Simplify each term.

Apply the distributive property.

$a = 8 - \frac{53}{4} + \frac{5 c}{4} - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Multiply $- - \frac{5 c}{4}$ .

Multiply $- 1$ by $- 1$ .

$a = 8 - \frac{53}{4} + 1 (\frac{5 c}{4}) - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Multiply $\frac{5 c}{4}$ by $1$ .

$a = 8 - \frac{53}{4} + \frac{5 c}{4} - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - \frac{53}{4} + \frac{5 c}{4} - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = 8 - \frac{53}{4} + \frac{5 c}{4} - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$a = 8 \cdot \frac{4}{4} - \frac{53}{4} + \frac{5 c}{4} - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Combine $8$ and $\frac{4}{4}$ .

$a = \frac{8 \cdot 4}{4} - \frac{53}{4} + \frac{5 c}{4} - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Combine the numerators over the common denominator.

$a = \frac{8 \cdot 4 - 53}{4} + \frac{5 c}{4} - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Simplify the numerator.

Multiply $8$ by $4$ .

$a = \frac{32 - 53}{4} + \frac{5 c}{4} - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Subtract $53$ from $32$ .

$a = \frac{- 21}{4} + \frac{5 c}{4} - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = \frac{- 21}{4} + \frac{5 c}{4} - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Move the negative in front of the fraction.

$a = - \frac{21}{4} + \frac{5 c}{4} - c$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

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

$a = - \frac{21}{4} + \frac{5 c}{4} - c \cdot \frac{4}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Combine $- c$ and $\frac{4}{4}$ .

$a = - \frac{21}{4} + \frac{5 c}{4} + \frac{- c \cdot 4}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Combine the numerators over the common denominator.

$a = - \frac{21}{4} + \frac{5 c - c \cdot 4}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Combine the numerators over the common denominator.

$a = \frac{- 21 + 5 c - c \cdot 4}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Multiply $4$ by $- 1$ .

$a = \frac{- 21 + 5 c - 4 c}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Subtract $4 c$ from $5 c$ .

$a = \frac{- 21 + c}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Rewrite $- 21$ as $- 1 (21)$ .

$a = \frac{- 1 \cdot 21 + c}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Factor $- 1$ out of $c$ .

$a = \frac{- 1 \cdot 21 - 1 (- c)}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Factor $- 1$ out of $- 1 (21) - 1 (- c)$ .

$a = \frac{- 1 (21 - c)}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Move the negative in front of the fraction.

$a = - \frac{21 - c}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = - \frac{21 - c}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = - \frac{21 - c}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = - \frac{21 - c}{4}$

$3 = \frac{11 - c}{2}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Solve for $c$ in $3 = \frac{11 - c}{2}$ .

Rewrite the equation as $\frac{11 - c}{2} = 3$ .

$\frac{11 - c}{2} = 3$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Multiply both sides by $2$ .

$\frac{11 - c}{2} \cdot 2 = 3 \cdot 2$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Simplify.

Simplify the left side.

Simplify $\frac{11 - c}{2} \cdot 2$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{11 - c}{2} \cdot 2 = 3 \cdot 2$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Rewrite the expression.

$11 - c = 3 \cdot 2$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$11 - c = 3 \cdot 2$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Reorder $11$ and $- c$ .

$- c + 11 = 3 \cdot 2$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$- c + 11 = 3 \cdot 2$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$- c + 11 = 3 \cdot 2$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Simplify the right side.

Multiply $3$ by $2$ .

$- c + 11 = 6$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$- c + 11 = 6$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$- c + 11 = 6$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Solve for $c$ .

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

Subtract $11$ from both sides of the equation.

$- c = 6 - 11$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Subtract $11$ from $6$ .

$- c = - 5$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$- c = - 5$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Divide each term in $- c = - 5$ by $- 1$ and simplify.

Divide each term in $- c = - 5$ by $- 1$ .

$\frac{- c}{- 1} = \frac{- 5}{- 1}$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{c}{1} = \frac{- 5}{- 1}$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Divide $c$ by $1$ .

$c = \frac{- 5}{- 1}$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$c = \frac{- 5}{- 1}$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Simplify the right side.

Divide $- 5$ by $- 1$ .

$c = 5$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$c = 5$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$c = 5$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$c = 5$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

$c = 5$

$a = - \frac{21 - c}{4}$

$b = \frac{53}{4} - \frac{5 c}{4}$

Replace all occurrences of $c$ with $5$ in each equation.

Replace all occurrences of $c$ in $a = - \frac{21 - c}{4}$ with $5$ .

$a = - \frac{21 - (5)}{4}$

$c = 5$

$b = \frac{53}{4} - \frac{5 c}{4}$

Simplify the right side.

Simplify $- \frac{21 - (5)}{4}$ .

Simplify the numerator.

Multiply $- 1$ by $5$ .

$a = - \frac{21 - 5}{4}$

$c = 5$

$b = \frac{53}{4} - \frac{5 c}{4}$

Subtract $5$ from $21$ .

$a = - \frac{16}{4}$

$c = 5$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = - \frac{16}{4}$

$c = 5$

$b = \frac{53}{4} - \frac{5 c}{4}$

Simplify the expression.

Divide $16$ by $4$ .

$a = - 1 \cdot 4$

$c = 5$

$b = \frac{53}{4} - \frac{5 c}{4}$

Multiply $- 1$ by $4$ .

$a = - 4$

$c = 5$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = - 4$

$c = 5$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = - 4$

$c = 5$

$b = \frac{53}{4} - \frac{5 c}{4}$

$a = - 4$

$c = 5$

$b = \frac{53}{4} - \frac{5 c}{4}$

Replace all occurrences of $c$ in $b = \frac{53}{4} - \frac{5 c}{4}$ with $5$ .

$b = \frac{53}{4} - \frac{5 (5)}{4}$

$a = - 4$

$c = 5$

Simplify the right side.

Simplify $\frac{53}{4} - \frac{5 (5)}{4}$ .

Combine the numerators over the common denominator.

$b = \frac{53 - 5 \cdot 5}{4}$

$a = - 4$

$c = 5$

Simplify the expression.

Multiply $- 5$ by $5$ .

$b = \frac{53 - 25}{4}$

$a = - 4$

$c = 5$

Subtract $25$ from $53$ .

$b = \frac{28}{4}$

$a = - 4$

$c = 5$

Divide $28$ by $4$ .

$b = 7$

$a = - 4$

$c = 5$

$b = 7$

$a = - 4$

$c = 5$

$b = 7$

$a = - 4$

$c = 5$

$b = 7$

$a = - 4$

$c = 5$

$b = 7$

$a = - 4$

$c = 5$

List all of the solutions.

$b = 7, a = - 4, c = 5$

Step 4

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

$y = - 4 x^{2} + 7 x + 5$

Step 5

Find the Equation of the Parabola (1,8) , (2,3) , (4,-31)

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