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

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

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.

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

Step 3

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

Solve for $a$ in $- 2 = a + b + c$ .

Rewrite the equation as $a + b + c = - 2$ .

$a + b + c = - 2$

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

$- 4 = a \cdot 3^{2} + b (3) + c$

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

Subtract $b$ from both sides of the equation.

$a + c = - 2 - b$

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

$- 4 = a \cdot 3^{2} + b (3) + c$

Subtract $c$ from both sides of the equation.

$a = - 2 - b - c$

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

$- 4 = a \cdot 3^{2} + b (3) + c$

$a = - 2 - b - c$

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

$- 4 = a \cdot 3^{2} + b (3) + c$

$a = - 2 - b - c$

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

$- 4 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $a$ with $- 2 - b - c$ in each equation.

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

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

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

Simplify the right side.

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

Simplify each term.

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

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

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

Apply the distributive property.

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

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

Simplify.

Multiply $- 2$ by $4$ .

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

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

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

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

Multiply $4$ by $- 1$ .

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

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

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

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

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

$- 4 = - 8 - 4 b - 4 c + 2 b + c$

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

$- 4 = - 8 - 4 b - 4 c + 2 b + c$

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

Simplify by adding terms.

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

$- 4 = - 8 - 2 b - 4 c + c$

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

Add $- 4 c$ and $c$ .

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 4 = a \cdot 3^{2} + b (3) + c$

Replace all occurrences of $a$ in $- 4 = a \cdot 3^{2} + b (3) + c$ with $- 2 - b - c$ .

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

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Simplify the right side.

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

Simplify each term.

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

$- 4 = (- 2 - b - c) \cdot 9 + b (3) + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Apply the distributive property.

$- 4 = - 2 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Simplify.

Multiply $- 2$ by $9$ .

$- 4 = - 18 - b \cdot 9 - c \cdot 9 + b (3) + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Multiply $9$ by $- 1$ .

$- 4 = - 18 - 9 b - c \cdot 9 + b (3) + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Multiply $9$ by $- 1$ .

$- 4 = - 18 - 9 b - 9 c + b (3) + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 4 = - 18 - 9 b - 9 c + b (3) + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

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

$- 4 = - 18 - 9 b - 9 c + 3 b + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 4 = - 18 - 9 b - 9 c + 3 b + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Simplify by adding terms.

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

$- 4 = - 18 - 6 b - 9 c + c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Add $- 9 c$ and $c$ .

$- 4 = - 18 - 6 b - 8 c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 4 = - 18 - 6 b - 8 c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 4 = - 18 - 6 b - 8 c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 4 = - 18 - 6 b - 8 c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 4 = - 18 - 6 b - 8 c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Solve for $b$ in $- 4 = - 18 - 6 b - 8 c$ .

Rewrite the equation as $- 18 - 6 b - 8 c = - 4$ .

$- 18 - 6 b - 8 c = - 4$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

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

Add $18$ to both sides of the equation.

$- 6 b - 8 c = - 4 + 18$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

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

$- 6 b = - 4 + 18 + 8 c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Add $- 4$ and $18$ .

$- 6 b = 14 + 8 c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$- 6 b = 14 + 8 c$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Divide each term in $- 6 b = 14 + 8 c$ by $- 6$ and simplify.

Divide each term in $- 6 b = 14 + 8 c$ by $- 6$ .

$\frac{- 6 b}{- 6} = \frac{14}{- 6} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 b}{- 6} = \frac{14}{- 6} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Divide $b$ by $1$ .

$b = \frac{14}{- 6} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = \frac{14}{- 6} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = \frac{14}{- 6} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Simplify the right side.

Simplify each term.

Cancel the common factor of $14$ and $- 6$ .

Factor $2$ out of $14$ .

$b = \frac{2 (7)}{- 6} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$b = \frac{2 \cdot 7}{2 \cdot - 3} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Cancel the common factor.

$b = \frac{2 \cdot 7}{2 \cdot - 3} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Rewrite the expression.

$b = \frac{7}{- 3} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = \frac{7}{- 3} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = \frac{7}{- 3} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Move the negative in front of the fraction.

$b = - \frac{7}{3} + \frac{8 c}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

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

Factor $2$ out of $8 c$ .

$b = - \frac{7}{3} + \frac{2 (4 c)}{- 6}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Cancel the common factors.

Factor $2$ out of $- 6$ .

$b = - \frac{7}{3} + \frac{2 (4 c)}{2 (- 3)}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Cancel the common factor.

$b = - \frac{7}{3} + \frac{2 (4 c)}{2 \cdot - 3}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Rewrite the expression.

$b = - \frac{7}{3} + \frac{4 c}{- 3}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{7}{3} + \frac{4 c}{- 3}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{7}{3} + \frac{4 c}{- 3}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Move the negative in front of the fraction.

$b = - \frac{7}{3} - \frac{4 c}{3}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$- 4 = - 8 - 2 b - 3 c$

$a = - 2 - b - c$

Replace all occurrences of $b$ with $- \frac{7}{3} - \frac{4 c}{3}$ in each equation.

Replace all occurrences of $b$ in $- 4 = - 8 - 2 b - 3 c$ with $- \frac{7}{3} - \frac{4 c}{3}$ .

$- 4 = - 8 - 2 (- \frac{7}{3} - \frac{4 c}{3}) - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Simplify the right side.

Simplify $- 8 - 2 (- \frac{7}{3} - \frac{4 c}{3}) - 3 c$ .

Simplify each term.

Apply the distributive property.

$- 4 = - 8 - 2 (- \frac{7}{3}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Multiply $- 2 (- \frac{7}{3})$ .

Multiply $- 1$ by $- 2$ .

$- 4 = - 8 + 2 (\frac{7}{3}) - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Combine $2$ and $\frac{7}{3}$ .

$- 4 = - 8 + \frac{2 \cdot 7}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Multiply $2$ by $7$ .

$- 4 = - 8 + \frac{14}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$- 4 = - 8 + \frac{14}{3} - 2 (- \frac{4 c}{3}) - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

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

Multiply $- 1$ by $- 2$ .

$- 4 = - 8 + \frac{14}{3} + 2 (\frac{4 c}{3}) - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

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

$- 4 = - 8 + \frac{14}{3} + \frac{2 (4 c)}{3} - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Multiply $4$ by $2$ .

$- 4 = - 8 + \frac{14}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$- 4 = - 8 + \frac{14}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$- 4 = - 8 + \frac{14}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 4 = - 8 \cdot \frac{3}{3} + \frac{14}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Combine $- 8$ and $\frac{3}{3}$ .

$- 4 = \frac{- 8 \cdot 3}{3} + \frac{14}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Combine the numerators over the common denominator.

$- 4 = \frac{- 8 \cdot 3 + 14}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Simplify the numerator.

Multiply $- 8$ by $3$ .

$- 4 = \frac{- 24 + 14}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Add $- 24$ and $14$ .

$- 4 = \frac{- 10}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$- 4 = \frac{- 10}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Move the negative in front of the fraction.

$- 4 = - \frac{10}{3} + \frac{8 c}{3} - 3 c$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

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

$- 4 = - \frac{10}{3} + \frac{8 c}{3} - 3 c \cdot \frac{3}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

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

$- 4 = - \frac{10}{3} + \frac{8 c}{3} + \frac{- 3 c \cdot 3}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Combine the numerators over the common denominator.

$- 4 = - \frac{10}{3} + \frac{8 c - 3 c \cdot 3}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Combine the numerators over the common denominator.

$- 4 = \frac{- 10 + 8 c - 3 c \cdot 3}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Multiply $3$ by $- 3$ .

$- 4 = \frac{- 10 + 8 c - 9 c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Subtract $9 c$ from $8 c$ .

$- 4 = \frac{- 10 - c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Rewrite $- 10$ as $- 1 (10)$ .

$- 4 = \frac{- 1 \cdot 10 - c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Factor $- 1$ out of $- c$ .

$- 4 = \frac{- 1 \cdot 10 - (c)}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Factor $- 1$ out of $- 1 (10) - (c)$ .

$- 4 = \frac{- 1 (10 + c)}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Move the negative in front of the fraction.

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 - b - c$

Replace all occurrences of $b$ in $a = - 2 - b - c$ with $- \frac{7}{3} - \frac{4 c}{3}$ .

$a = - 2 - (- \frac{7}{3} - \frac{4 c}{3}) - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Simplify the right side.

Simplify $- 2 - (- \frac{7}{3} - \frac{4 c}{3}) - c$ .

Simplify each term.

Apply the distributive property.

$a = - 2 + \frac{7}{3} + \frac{4 c}{3} - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Multiply $- - \frac{7}{3}$ .

Multiply $- 1$ by $- 1$ .

$a = - 2 + 1 (\frac{7}{3}) + \frac{4 c}{3} - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Multiply $\frac{7}{3}$ by $1$ .

$a = - 2 + \frac{7}{3} + \frac{4 c}{3} - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 + \frac{7}{3} + \frac{4 c}{3} - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

Multiply $- 1$ by $- 1$ .

$a = - 2 + \frac{7}{3} + 1 (\frac{4 c}{3}) - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

$a = - 2 + \frac{7}{3} + \frac{4 c}{3} - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 + \frac{7}{3} + \frac{4 c}{3} - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = - 2 + \frac{7}{3} + \frac{4 c}{3} - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

$a = - 2 \cdot \frac{3}{3} + \frac{7}{3} + \frac{4 c}{3} - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

$a = \frac{- 2 \cdot 3}{3} + \frac{7}{3} + \frac{4 c}{3} - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{- 2 \cdot 3 + 7}{3} + \frac{4 c}{3} - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Simplify the numerator.

Multiply $- 2$ by $3$ .

$a = \frac{- 6 + 7}{3} + \frac{4 c}{3} - c$

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Add $- 6$ and $7$ .

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

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

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

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

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

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

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

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Combine the numerators over the common denominator.

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

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Multiply $3$ by $- 1$ .

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

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Subtract $3 c$ from $4 c$ .

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

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

$- 4 = - \frac{10 + c}{3}$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Solve for $c$ in $- 4 = - \frac{10 + c}{3}$ .

Rewrite the equation as $- \frac{10 + c}{3} = - 4$ .

$- \frac{10 + c}{3} = - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

$- 3 (- \frac{10 + c}{3}) = - 3 \cdot - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 3 (- \frac{10 + c}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{10 + c}{3}$ into the numerator.

$- 3 \frac{- (10 + c)}{3} = - 3 \cdot - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

Factor $3$ out of $- 3$ .

$3 (- 1) (\frac{- (10 + c)}{3}) = - 3 \cdot - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

Cancel the common factor.

$3 \cdot (- 1 \frac{- (10 + c)}{3}) = - 3 \cdot - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

Rewrite the expression.

$10 + c = - 3 \cdot - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

$10 + c = - 3 \cdot - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (10 + c) = - 3 \cdot - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

Multiply $10 + c$ by $1$ .

$10 + c = - 3 \cdot - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

$10 + c = - 3 \cdot - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

$10 + c = - 3 \cdot - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

$10 + c = - 3 \cdot - 4$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

Simplify the right side.

Multiply $- 3$ by $- 4$ .

$10 + c = 12$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

$10 + c = 12$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

$10 + c = 12$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

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

Subtract $10$ from both sides of the equation.

$c = 12 - 10$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

Subtract $10$ from $12$ .

$c = 2$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

$c = 2$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

$c = 2$

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

$b = - \frac{7}{3} - \frac{4 c}{3}$

Replace all occurrences of $c$ with $2$ in each equation.

Replace all occurrences of $c$ in $a = \frac{1 + c}{3}$ with $2$ .

$a = \frac{1 + 2}{3}$

$c = 2$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Simplify $a = \frac{1 + 2}{3}$ .

Simplify the left side.

Remove parentheses.

$a = \frac{1 + 2}{3}$

$c = 2$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = \frac{1 + 2}{3}$

$c = 2$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Simplify the right side.

Simplify $\frac{1 + 2}{3}$ .

Add $1$ and $2$ .

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

$c = 2$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Divide $3$ by $3$ .

$a = 1$

$c = 2$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = 1$

$c = 2$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = 1$

$c = 2$

$b = - \frac{7}{3} - \frac{4 c}{3}$

$a = 1$

$c = 2$

$b = - \frac{7}{3} - \frac{4 c}{3}$

Replace all occurrences of $c$ in $b = - \frac{7}{3} - \frac{4 c}{3}$ with $2$ .

$b = - \frac{7}{3} - \frac{4 (2)}{3}$

$a = 1$

$c = 2$

Simplify the right side.

Simplify $- \frac{7}{3} - \frac{4 (2)}{3}$ .

Combine the numerators over the common denominator.

$b = \frac{- 7 - 4 \cdot 2}{3}$

$a = 1$

$c = 2$

Simplify the expression.

Multiply $- 4$ by $2$ .

$b = \frac{- 7 - 8}{3}$

$a = 1$

$c = 2$

Subtract $8$ from $- 7$ .

$b = \frac{- 15}{3}$

$a = 1$

$c = 2$

Divide $- 15$ by $3$ .

$b = - 5$

$a = 1$

$c = 2$

$b = - 5$

$a = 1$

$c = 2$

$b = - 5$

$a = 1$

$c = 2$

$b = - 5$

$a = 1$

$c = 2$

$b = - 5$

$a = 1$

$c = 2$

List all of the solutions.

$b = - 5, a = 1, c = 2$

Step 4

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

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

Step 5

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

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