Find the Equation of the Parabola (1,-8) , (2,-4) , (3,6)

$(1, - 8)$ , $(2, - 4)$ , $(3, 6)$

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.

$- 8 = a {(1)}^{2} + b (1) + c, - 4 = a {(2)}^{2} + b (2) + c, 6 = a {(3)}^{2} + b (3) + c$

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 $a + b + c = - 8$ .

$a + b + c = - 8$

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

$6 = a \cdot 3^{2} + b (3) + c$

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

Subtract $b$ from both sides of the equation.

$a + c = - 8 - b$

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

$6 = a \cdot 3^{2} + b (3) + c$

Subtract $c$ from both sides of the equation.

$a = - 8 - b - c$

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

$6 = a \cdot 3^{2} + b (3) + c$

$a = - 8 - b - c$

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

$6 = a \cdot 3^{2} + b (3) + c$

$a = - 8 - b - c$

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

$6 = a \cdot 3^{2} + b (3) + c$

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

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

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

Simplify the right side.

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

Simplify each term.

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

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

Apply the distributive property.

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

Simplify.

Multiply $- 8$ by $4$ .

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

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

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

Simplify by adding terms.

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

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

Add $- 4 c$ and $c$ .

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

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

$a = - 8 - b - c$

$6 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $a$ in $6 = a \cdot 3^{2} + b (3) + c$ with $- 8 - b - c$ .

$6 = (- 8 - b - c) \cdot 3^{2} + b (3) + c$

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

$a = - 8 - b - c$

Simplify the right side.

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

Simplify each term.

Raise $3$ to the power of $2$ .

$6 = (- 8 - b - c) \cdot 9 + b (3) + c$

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

$a = - 8 - b - c$

Apply the distributive property.

$6 = - 8 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

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

$a = - 8 - b - c$

Simplify.

Multiply $- 8$ by $9$ .

$6 = - 72 - b \cdot 9 - c \cdot 9 + b (3) + c$

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

$a = - 8 - b - c$

Multiply $9$ by $- 1$ .

$6 = - 72 - 9 b - c \cdot 9 + b (3) + c$

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

$a = - 8 - b - c$

Multiply $9$ by $- 1$ .

$6 = - 72 - 9 b - 9 c + b (3) + c$

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

$a = - 8 - b - c$

$6 = - 72 - 9 b - 9 c + b (3) + c$

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

$a = - 8 - b - c$

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

$6 = - 72 - 9 b - 9 c + 3 b + c$

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

$a = - 8 - b - c$

$6 = - 72 - 9 b - 9 c + 3 b + c$

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

$a = - 8 - b - c$

Simplify by adding terms.

Add $- 9 b$ and $3 b$ .

$6 = - 72 - 6 b - 9 c + c$

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

$a = - 8 - b - c$

Add $- 9 c$ and $c$ .

$6 = - 72 - 6 b - 8 c$

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

$a = - 8 - b - c$

$6 = - 72 - 6 b - 8 c$

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

$a = - 8 - b - c$

$6 = - 72 - 6 b - 8 c$

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

$a = - 8 - b - c$

$6 = - 72 - 6 b - 8 c$

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

$a = - 8 - b - c$

$6 = - 72 - 6 b - 8 c$

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

$a = - 8 - b - c$

Solve for $b$ in $6 = - 72 - 6 b - 8 c$ .

Rewrite the equation as $- 72 - 6 b - 8 c = 6$ .

$- 72 - 6 b - 8 c = 6$

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

$a = - 8 - b - c$

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

Add $72$ to both sides of the equation.

$- 6 b - 8 c = 6 + 72$

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

$a = - 8 - b - c$

Add $8 c$ to both sides of the equation.

$- 6 b = 6 + 72 + 8 c$

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

$a = - 8 - b - c$

Add $6$ and $72$ .

$- 6 b = 78 + 8 c$

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

$a = - 8 - b - c$

$- 6 b = 78 + 8 c$

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

$a = - 8 - b - c$

Divide each term in $- 6 b = 78 + 8 c$ by $- 6$ and simplify.

Divide each term in $- 6 b = 78 + 8 c$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{78}{- 6} + \frac{8 c}{- 6}$

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

$a = - 8 - b - c$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{78}{- 6} + \frac{8 c}{- 6}$

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

$a = - 8 - b - c$

Divide $b$ by $1$ .

$b = \frac{78}{- 6} + \frac{8 c}{- 6}$

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

$a = - 8 - b - c$

$b = \frac{78}{- 6} + \frac{8 c}{- 6}$

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

$a = - 8 - b - c$

$b = \frac{78}{- 6} + \frac{8 c}{- 6}$

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

$a = - 8 - b - c$

Simplify the right side.

Simplify each term.

Divide $78$ by $- 6$ .

$b = - 13 + \frac{8 c}{- 6}$

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

$a = - 8 - b - c$

Cancel the common factor of $8$ and $- 6$ .

Factor $2$ out of $8 c$ .

$b = - 13 + \frac{2 (4 c)}{- 6}$

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

$a = - 8 - b - c$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$b = - 13 + \frac{2 (4 c)}{2 (- 3)}$

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

$a = - 8 - b - c$

Cancel the common factor.

$b = - 13 + \frac{2 (4 c)}{2 \cdot - 3}$

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

$a = - 8 - b - c$

Rewrite the expression.

$b = - 13 + \frac{4 c}{- 3}$

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

$a = - 8 - b - c$

$b = - 13 + \frac{4 c}{- 3}$

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

$a = - 8 - b - c$

$b = - 13 + \frac{4 c}{- 3}$

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

$a = - 8 - b - c$

Move the negative in front of the fraction.

$b = - 13 - \frac{4 c}{3}$

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

$a = - 8 - b - c$

$b = - 13 - \frac{4 c}{3}$

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

$a = - 8 - b - c$

$b = - 13 - \frac{4 c}{3}$

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

$a = - 8 - b - c$

$b = - 13 - \frac{4 c}{3}$

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

$a = - 8 - b - c$

$b = - 13 - \frac{4 c}{3}$

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

$a = - 8 - b - c$

Replace all occurrences of $b$ with $- 13 - \frac{4 c}{3}$ in each equation.

Replace all occurrences of $b$ in $- 4 = - 32 - 2 b - 3 c$ with $- 13 - \frac{4 c}{3}$ .

$- 4 = - 32 - 2 (- 13 - \frac{4 c}{3}) - 3 c$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Simplify the right side.

Simplify $- 32 - 2 (- 13 - \frac{4 c}{3}) - 3 c$ .

Simplify each term.

Apply the distributive property.

$- 4 = - 32 - 2 \cdot - 13 - 2 (- \frac{4 c}{3}) - 3 c$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Multiply $- 2$ by $- 13$ .

$- 4 = - 32 + 26 - 2 (- \frac{4 c}{3}) - 3 c$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Multiply $- 2 (- \frac{4 c}{3})$ .

Multiply $- 1$ by $- 2$ .

$- 4 = - 32 + 26 + 2 (\frac{4 c}{3}) - 3 c$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Combine $2$ and $\frac{4 c}{3}$ .

$- 4 = - 32 + 26 + \frac{2 (4 c)}{3} - 3 c$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Multiply $4$ by $2$ .

$- 4 = - 32 + 26 + \frac{8 c}{3} - 3 c$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

$- 4 = - 32 + 26 + \frac{8 c}{3} - 3 c$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

$- 4 = - 32 + 26 + \frac{8 c}{3} - 3 c$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Add $- 32$ and $26$ .

$- 4 = - 6 + \frac{8 c}{3} - 3 c$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

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

$- 4 = - 6 + \frac{8 c}{3} - 3 c \cdot \frac{3}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Simplify terms.

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

$- 4 = - 6 + \frac{8 c}{3} + \frac{- 3 c \cdot 3}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Combine the numerators over the common denominator.

$- 4 = - 6 + \frac{8 c - 3 c \cdot 3}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

$- 4 = - 6 + \frac{8 c - 3 c \cdot 3}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Simplify each term.

Simplify the numerator.

Factor $c$ out of $8 c - 3 c \cdot 3$ .

Factor $c$ out of $8 c$ .

$- 4 = - 6 + \frac{c \cdot 8 - 3 c \cdot 3}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Factor $c$ out of $- 3 c \cdot 3$ .

$- 4 = - 6 + \frac{c \cdot 8 + c (- 3 \cdot 3)}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Factor $c$ out of $c \cdot 8 + c (- 3 \cdot 3)$ .

$- 4 = - 6 + \frac{c (8 - 3 \cdot 3)}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

$- 4 = - 6 + \frac{c (8 - 3 \cdot 3)}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Multiply $- 3$ by $3$ .

$- 4 = - 6 + \frac{c (8 - 9)}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Subtract $9$ from $8$ .

$- 4 = - 6 + \frac{c \cdot - 1}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

$- 4 = - 6 + \frac{c \cdot - 1}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Move $- 1$ to the left of $c$ .

$- 4 = - 6 + \frac{- 1 \cdot c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Move the negative in front of the fraction.

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 - b - c$

Replace all occurrences of $b$ in $a = - 8 - b - c$ with $- 13 - \frac{4 c}{3}$ .

$a = - 8 - (- 13 - \frac{4 c}{3}) - c$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Simplify the right side.

Simplify $- 8 - (- 13 - \frac{4 c}{3}) - c$ .

Simplify each term.

Apply the distributive property.

$a = - 8 + 13 + \frac{4 c}{3} - c$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Multiply $- 1$ by $- 13$ .

$a = - 8 + 13 + \frac{4 c}{3} - c$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Multiply $- - \frac{4 c}{3}$ .

Multiply $- 1$ by $- 1$ .

$a = - 8 + 13 + 1 (\frac{4 c}{3}) - c$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Multiply $\frac{4 c}{3}$ by $1$ .

$a = - 8 + 13 + \frac{4 c}{3} - c$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 + 13 + \frac{4 c}{3} - c$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = - 8 + 13 + \frac{4 c}{3} - c$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Add $- 8$ and $13$ .

$a = 5 + \frac{4 c}{3} - c$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

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

$a = 5 + \frac{4 c}{3} - c \cdot \frac{3}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Simplify terms.

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

$a = 5 + \frac{4 c}{3} + \frac{- c \cdot 3}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = 5 + \frac{4 c - c \cdot 3}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = 5 + \frac{4 c - c \cdot 3}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Simplify each term.

Simplify the numerator.

Factor $c$ out of $4 c - c \cdot 3$ .

Factor $c$ out of $4 c$ .

$a = 5 + \frac{c \cdot 4 - c \cdot 3}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Factor $c$ out of $- c \cdot 3$ .

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

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Factor $c$ out of $c \cdot 4 + c (- 1 \cdot 3)$ .

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

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

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

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Multiply $- 1$ by $3$ .

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

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Subtract $3$ from $4$ .

$a = 5 + \frac{c \cdot 1}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = 5 + \frac{c \cdot 1}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Multiply $c$ by $1$ .

$a = 5 + \frac{c}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = 5 + \frac{c}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = 5 + \frac{c}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = 5 + \frac{c}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$a = 5 + \frac{c}{3}$

$- 4 = - 6 - \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Solve for $c$ in $- 4 = - 6 - \frac{c}{3}$ .

Rewrite the equation as $- 6 - \frac{c}{3} = - 4$ .

$- 6 - \frac{c}{3} = - 4$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

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

Add $6$ to both sides of the equation.

$- \frac{c}{3} = - 4 + 6$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Add $- 4$ and $6$ .

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

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

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

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{c}{3}) = - 3 \cdot 2$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 3 (- \frac{c}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{c}{3}$ into the numerator.

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

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- c}{3}) = - 3 \cdot 2$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Cancel the common factor.

$3 \cdot (- 1 \frac{- c}{3}) = - 3 \cdot 2$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Rewrite the expression.

$c = - 3 \cdot 2$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$c = - 3 \cdot 2$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 c = - 3 \cdot 2$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Multiply $c$ by $1$ .

$c = - 3 \cdot 2$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$c = - 3 \cdot 2$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$c = - 3 \cdot 2$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$c = - 3 \cdot 2$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Simplify the right side.

Multiply $- 3$ by $2$ .

$c = - 6$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$c = - 6$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$c = - 6$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

$c = - 6$

$a = 5 + \frac{c}{3}$

$b = - 13 - \frac{4 c}{3}$

Replace all occurrences of $c$ with $- 6$ in each equation.

Replace all occurrences of $c$ in $a = 5 + \frac{c}{3}$ with $- 6$ .

$a = 5 + \frac{- 6}{3}$

$c = - 6$

$b = - 13 - \frac{4 c}{3}$

Simplify the right side.

Simplify $5 + \frac{- 6}{3}$ .

Divide $- 6$ by $3$ .

$a = 5 - 2$

$c = - 6$

$b = - 13 - \frac{4 c}{3}$

Subtract $2$ from $5$ .

$a = 3$

$c = - 6$

$b = - 13 - \frac{4 c}{3}$

$a = 3$

$c = - 6$

$b = - 13 - \frac{4 c}{3}$

$a = 3$

$c = - 6$

$b = - 13 - \frac{4 c}{3}$

Replace all occurrences of $c$ in $b = - 13 - \frac{4 c}{3}$ with $- 6$ .

$b = - 13 - \frac{4 (- 6)}{3}$

$a = 3$

$c = - 6$

Simplify the right side.

Simplify $- 13 - \frac{4 (- 6)}{3}$ .

Simplify each term.

Cancel the common factor of $- 6$ and $3$ .

Factor $3$ out of $4 (- 6)$ .

$b = - 13 - \frac{3 (4 \cdot (- 2))}{3}$

$a = 3$

$c = - 6$

Cancel the common factors.

Factor $3$ out of $3$ .

$b = - 13 - \frac{3 (4 \cdot (- 2))}{3 (1)}$

$a = 3$

$c = - 6$

Cancel the common factor.

$b = - 13 - \frac{3 (4 \cdot (- 2))}{3 \cdot 1}$

$a = 3$

$c = - 6$

Rewrite the expression.

$b = - 13 - \frac{4 \cdot (- 2)}{1}$

$a = 3$

$c = - 6$

Divide $4 \cdot (- 2)$ by $1$ .

$b = - 13 - (4 \cdot (- 2))$

$a = 3$

$c = - 6$

$b = - 13 - (4 \cdot (- 2))$

$a = 3$

$c = - 6$

$b = - 13 - (4 \cdot (- 2))$

$a = 3$

$c = - 6$

Multiply $- (4 \cdot (- 2))$ .

Multiply $4$ by $- 2$ .

$b = - 13 + 8$

$a = 3$

$c = - 6$

Multiply $- 1$ by $- 8$ .

$b = - 13 + 8$

$a = 3$

$c = - 6$

$b = - 13 + 8$

$a = 3$

$c = - 6$

$b = - 13 + 8$

$a = 3$

$c = - 6$

Add $- 13$ and $8$ .

$b = - 5$

$a = 3$

$c = - 6$

$b = - 5$

$a = 3$

$c = - 6$

$b = - 5$

$a = 3$

$c = - 6$

$b = - 5$

$a = 3$

$c = - 6$

List all of the solutions.

$b = - 5, a = 3, c = - 6$

Step 4

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

$y = 3 x^{2} - 5 x - 6$

Step 5

Find the Equation of the Parabola (1,-8) , (2,-4) , (3,6)

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