Find the Equation of the Parabola (1,3) , (2,-7) , (-3,11)

$(1, 3)$ , $(2, - 7)$ , $(- 3, 11)$

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.

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

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.

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

$- 7 = a {(2)}^{2} + b (2) + c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

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

Solve for $a$ in $3 = a + b + c$ .

Rewrite the equation as $a + b + c = 3$ .

$a + b + c = 3$

$- 7 = a \cdot 2^{2} + b (2) + c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

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

Subtract $b$ from both sides of the equation.

$a + c = 3 - b$

$- 7 = a \cdot 2^{2} + b (2) + c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

Subtract $c$ from both sides of the equation.

$a = 3 - b - c$

$- 7 = a \cdot 2^{2} + b (2) + c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

$a = 3 - b - c$

$- 7 = a \cdot 2^{2} + b (2) + c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

$a = 3 - b - c$

$- 7 = a \cdot 2^{2} + b (2) + c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

Replace all occurrences of $a$ with $3 - b - c$ in each equation.

Replace all occurrences of $a$ in $- 7 = a \cdot 2^{2} + b (2) + c$ with $3 - b - c$ .

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

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

Simplify the right side.

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

Simplify each term.

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

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

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

Apply the distributive property.

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

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

Simplify.

Multiply $3$ by $4$ .

$- 7 = 12 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

Multiply $4$ by $- 1$ .

$- 7 = 12 - 4 b - c \cdot 4 + b (2) + c$

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

Multiply $4$ by $- 1$ .

$- 7 = 12 - 4 b - 4 c + b (2) + c$

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

$- 7 = 12 - 4 b - 4 c + b (2) + c$

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

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

$- 7 = 12 - 4 b - 4 c + 2 b + c$

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

$- 7 = 12 - 4 b - 4 c + 2 b + c$

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

Simplify by adding terms.

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

$- 7 = 12 - 2 b - 4 c + c$

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

Add $- 4 c$ and $c$ .

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$11 = a {(- 3)}^{2} + b (- 3) + c$

Replace all occurrences of $a$ in $11 = a {(- 3)}^{2} + b (- 3) + c$ with $3 - b - c$ .

$11 = (3 - b - c) {(- 3)}^{2} + b (- 3) + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Simplify the right side.

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

Simplify each term.

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

$11 = (3 - b - c) \cdot 9 + b (- 3) + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Apply the distributive property.

$11 = 3 \cdot 9 - b \cdot 9 - c \cdot 9 + b (- 3) + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Simplify.

Multiply $3$ by $9$ .

$11 = 27 - b \cdot 9 - c \cdot 9 + b (- 3) + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Multiply $9$ by $- 1$ .

$11 = 27 - 9 b - c \cdot 9 + b (- 3) + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Multiply $9$ by $- 1$ .

$11 = 27 - 9 b - 9 c + b (- 3) + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$11 = 27 - 9 b - 9 c + b (- 3) + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

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

$11 = 27 - 9 b - 9 c - 3 b + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$11 = 27 - 9 b - 9 c - 3 b + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Simplify by adding terms.

Subtract $3 b$ from $- 9 b$ .

$11 = 27 - 12 b - 9 c + c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Add $- 9 c$ and $c$ .

$11 = 27 - 12 b - 8 c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$11 = 27 - 12 b - 8 c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$11 = 27 - 12 b - 8 c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$11 = 27 - 12 b - 8 c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$11 = 27 - 12 b - 8 c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Solve for $b$ in $11 = 27 - 12 b - 8 c$ .

Rewrite the equation as $27 - 12 b - 8 c = 11$ .

$27 - 12 b - 8 c = 11$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

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

Subtract $27$ from both sides of the equation.

$- 12 b - 8 c = 11 - 27$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

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

$- 12 b = 11 - 27 + 8 c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Subtract $27$ from $11$ .

$- 12 b = - 16 + 8 c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$- 12 b = - 16 + 8 c$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Divide each term in $- 12 b = - 16 + 8 c$ by $- 12$ and simplify.

Divide each term in $- 12 b = - 16 + 8 c$ by $- 12$ .

$\frac{- 12 b}{- 12} = \frac{- 16}{- 12} + \frac{8 c}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 b}{- 12} = \frac{- 16}{- 12} + \frac{8 c}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Divide $b$ by $1$ .

$b = \frac{- 16}{- 12} + \frac{8 c}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$b = \frac{- 16}{- 12} + \frac{8 c}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$b = \frac{- 16}{- 12} + \frac{8 c}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 16$ and $- 12$ .

Factor $- 4$ out of $- 16$ .

$b = \frac{- 4 \cdot 4}{- 12} + \frac{8 c}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Cancel the common factors.

Factor $- 4$ out of $- 12$ .

$b = \frac{- 4 \cdot 4}{- 4 \cdot 3} + \frac{8 c}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Cancel the common factor.

$b = \frac{- 4 \cdot 4}{- 4 \cdot 3} + \frac{8 c}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Rewrite the expression.

$b = \frac{4}{3} + \frac{8 c}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$b = \frac{4}{3} + \frac{8 c}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$b = \frac{4}{3} + \frac{8 c}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Cancel the common factor of $8$ and $- 12$ .

Factor $4$ out of $8 c$ .

$b = \frac{4}{3} + \frac{4 (2 c)}{- 12}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Cancel the common factors.

Factor $4$ out of $- 12$ .

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

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Cancel the common factor.

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

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Rewrite the expression.

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

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

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

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

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

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

Move the negative in front of the fraction.

$b = \frac{4}{3} - \frac{2 c}{3}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$- 7 = 12 - 2 b - 3 c$

$a = 3 - b - c$

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

Replace all occurrences of $b$ in $- 7 = 12 - 2 b - 3 c$ with $\frac{4}{3} - \frac{2 c}{3}$ .

$- 7 = 12 - 2 (\frac{4}{3} - \frac{2 c}{3}) - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Simplify the right side.

Simplify $12 - 2 (\frac{4}{3} - \frac{2 c}{3}) - 3 c$ .

Simplify each term.

Apply the distributive property.

$- 7 = 12 - 2 (\frac{4}{3}) - 2 (- \frac{2 c}{3}) - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

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

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

$- 7 = 12 + \frac{- 2 \cdot 4}{3} - 2 (- \frac{2 c}{3}) - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Multiply $- 2$ by $4$ .

$- 7 = 12 + \frac{- 8}{3} - 2 (- \frac{2 c}{3}) - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

$- 7 = 12 + \frac{- 8}{3} - 2 (- \frac{2 c}{3}) - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

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

Multiply $- 1$ by $- 2$ .

$- 7 = 12 + \frac{- 8}{3} + 2 (\frac{2 c}{3}) - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

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

$- 7 = 12 + \frac{- 8}{3} + \frac{2 (2 c)}{3} - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Multiply $2$ by $2$ .

$- 7 = 12 + \frac{- 8}{3} + \frac{4 c}{3} - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

$- 7 = 12 + \frac{- 8}{3} + \frac{4 c}{3} - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Move the negative in front of the fraction.

$- 7 = 12 - \frac{8}{3} + \frac{4 c}{3} - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

$- 7 = 12 - \frac{8}{3} + \frac{4 c}{3} - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

To write $12$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 7 = 12 \cdot \frac{3}{3} - \frac{8}{3} + \frac{4 c}{3} - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Combine $12$ and $\frac{3}{3}$ .

$- 7 = \frac{12 \cdot 3}{3} - \frac{8}{3} + \frac{4 c}{3} - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Combine the numerators over the common denominator.

$- 7 = \frac{12 \cdot 3 - 8}{3} + \frac{4 c}{3} - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Simplify the numerator.

Multiply $12$ by $3$ .

$- 7 = \frac{36 - 8}{3} + \frac{4 c}{3} - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Subtract $8$ from $36$ .

$- 7 = \frac{28}{3} + \frac{4 c}{3} - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

$- 7 = \frac{28}{3} + \frac{4 c}{3} - 3 c$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

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

$- 7 = \frac{28}{3} + \frac{4 c}{3} - 3 c \cdot \frac{3}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

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

$- 7 = \frac{28}{3} + \frac{4 c}{3} + \frac{- 3 c \cdot 3}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Combine the numerators over the common denominator.

$- 7 = \frac{28}{3} + \frac{4 c - 3 c \cdot 3}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Combine the numerators over the common denominator.

$- 7 = \frac{28 + 4 c - 3 c \cdot 3}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Multiply $3$ by $- 3$ .

$- 7 = \frac{28 + 4 c - 9 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

Subtract $9 c$ from $4 c$ .

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = 3 - b - c$

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

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Simplify the right side.

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

Simplify each term.

Apply the distributive property.

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Multiply $- - \frac{2 c}{3}$ .

Multiply $- 1$ by $- 1$ .

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Multiply $\frac{2 c}{3}$ by $1$ .

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

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

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

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

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Combine the numerators over the common denominator.

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Simplify the numerator.

Multiply $3$ by $3$ .

$a = \frac{9 - 4}{3} + \frac{2 c}{3} - c$

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Subtract $4$ from $9$ .

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

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

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

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

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Combine the numerators over the common denominator.

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Combine the numerators over the common denominator.

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Multiply $3$ by $- 1$ .

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Subtract $3 c$ from $2 c$ .

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

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

$- 7 = \frac{28 - 5 c}{3}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Solve for $c$ in $- 7 = \frac{28 - 5 c}{3}$ .

Rewrite the equation as $\frac{28 - 5 c}{3} = - 7$ .

$\frac{28 - 5 c}{3} = - 7$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Multiply both sides by $3$ .

$\frac{28 - 5 c}{3} \cdot 3 = - 7 \cdot 3$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Simplify.

Simplify the left side.

Simplify $\frac{28 - 5 c}{3} \cdot 3$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{28 - 5 c}{3} \cdot 3 = - 7 \cdot 3$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Rewrite the expression.

$28 - 5 c = - 7 \cdot 3$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$28 - 5 c = - 7 \cdot 3$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Reorder $28$ and $- 5 c$ .

$- 5 c + 28 = - 7 \cdot 3$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$- 5 c + 28 = - 7 \cdot 3$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$- 5 c + 28 = - 7 \cdot 3$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Simplify the right side.

Multiply $- 7$ by $3$ .

$- 5 c + 28 = - 21$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$- 5 c + 28 = - 21$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$- 5 c + 28 = - 21$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Solve for $c$ .

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

Subtract $28$ from both sides of the equation.

$- 5 c = - 21 - 28$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Subtract $28$ from $- 21$ .

$- 5 c = - 49$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$- 5 c = - 49$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Divide each term in $- 5 c = - 49$ by $- 5$ and simplify.

Divide each term in $- 5 c = - 49$ by $- 5$ .

$\frac{- 5 c}{- 5} = \frac{- 49}{- 5}$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 c}{- 5} = \frac{- 49}{- 5}$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Divide $c$ by $1$ .

$c = \frac{- 49}{- 5}$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$c = \frac{- 49}{- 5}$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$c = \frac{- 49}{- 5}$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Simplify the right side.

Dividing two negative values results in a positive value.

$c = \frac{49}{5}$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$c = \frac{49}{5}$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$c = \frac{49}{5}$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$c = \frac{49}{5}$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

$c = \frac{49}{5}$

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

$b = \frac{4}{3} - \frac{2 c}{3}$

Replace all occurrences of $c$ with $\frac{49}{5}$ in each equation.

Replace all occurrences of $c$ in $a = \frac{5 - c}{3}$ with $\frac{49}{5}$ .

$a = \frac{5 - (\frac{49}{5})}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Simplify the right side.

Simplify $\frac{5 - (\frac{49}{5})}{3}$ .

Simplify the numerator.

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

$a = \frac{5 \cdot \frac{5}{5} - \frac{49}{5}}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Combine $5$ and $\frac{5}{5}$ .

$a = \frac{\frac{5 \cdot 5}{5} - \frac{49}{5}}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{\frac{5 \cdot 5 - 49}{5}}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Simplify the numerator.

Multiply $5$ by $5$ .

$a = \frac{\frac{25 - 49}{5}}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Subtract $49$ from $25$ .

$a = \frac{\frac{- 24}{5}}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = \frac{\frac{- 24}{5}}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Move the negative in front of the fraction.

$a = \frac{- \frac{24}{5}}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = \frac{- \frac{24}{5}}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Multiply the numerator by the reciprocal of the denominator.

$a = - \frac{24}{5} \cdot \frac{1}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{24}{5}$ into the numerator.

$a = \frac{- 24}{5} \cdot \frac{1}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Factor $3$ out of $- 24$ .

$a = \frac{3 (- 8)}{5} \cdot \frac{1}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Cancel the common factor.

$a = \frac{3 \cdot - 8}{5} \cdot \frac{1}{3}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Rewrite the expression.

$a = \frac{- 8}{5}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = \frac{- 8}{5}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Move the negative in front of the fraction.

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

$b = \frac{4}{3} - \frac{2 c}{3}$

Replace all occurrences of $c$ in $b = \frac{4}{3} - \frac{2 c}{3}$ with $\frac{49}{5}$ .

$b = \frac{4}{3} - \frac{2 (\frac{49}{5})}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Simplify the right side.

Simplify $\frac{4}{3} - \frac{2 (\frac{49}{5})}{3}$ .

Combine the numerators over the common denominator.

$b = \frac{4 - 2 (\frac{49}{5})}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Simplify each term.

Multiply $- 2 (\frac{49}{5})$ .

Combine $- 2$ and $\frac{49}{5}$ .

$b = \frac{4 + \frac{- 2 \cdot 49}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Multiply $- 2$ by $49$ .

$b = \frac{4 + \frac{- 98}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

$b = \frac{4 + \frac{- 98}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Move the negative in front of the fraction.

$b = \frac{4 - \frac{98}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

$b = \frac{4 - \frac{98}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$b = \frac{4 \cdot \frac{5}{5} - \frac{98}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Combine $4$ and $\frac{5}{5}$ .

$b = \frac{\frac{4 \cdot 5}{5} - \frac{98}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Combine the numerators over the common denominator.

$b = \frac{\frac{4 \cdot 5 - 98}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Simplify the numerator.

Multiply $4$ by $5$ .

$b = \frac{\frac{20 - 98}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Subtract $98$ from $20$ .

$b = \frac{\frac{- 78}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

$b = \frac{\frac{- 78}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Move the negative in front of the fraction.

$b = \frac{- \frac{78}{5}}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Multiply the numerator by the reciprocal of the denominator.

$b = - \frac{78}{5} \cdot \frac{1}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{78}{5}$ into the numerator.

$b = \frac{- 78}{5} \cdot \frac{1}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Factor $3$ out of $- 78$ .

$b = \frac{3 (- 26)}{5} \cdot \frac{1}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Cancel the common factor.

$b = \frac{3 \cdot - 26}{5} \cdot \frac{1}{3}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Rewrite the expression.

$b = \frac{- 26}{5}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

$b = \frac{- 26}{5}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

Move the negative in front of the fraction.

$b = - \frac{26}{5}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

$b = - \frac{26}{5}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

$b = - \frac{26}{5}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

$b = - \frac{26}{5}$

$a = - \frac{8}{5}$

$c = \frac{49}{5}$

List all of the solutions.

$b = - \frac{26}{5}, a = - \frac{8}{5}, c = \frac{49}{5}$

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

$y = - \frac{8 x^{2}}{5} - \frac{26 x}{5} + \frac{49}{5}$

Find the Equation of the Parabola (1,3) , (2,-7) , (-3,11)

Related Questions

Support LCMGCF.com by disabling your adblocker.