Find the Equation of the Parabola (5,210) , (1,14) , (-3,10)

$(5, 210)$ , $(1, 14)$ , $(- 3, 10)$

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.

$210 = a {(5)}^{2} + b (5) + c, 14 = a {(1)}^{2} + b (1) + c, 10 = a {(- 3)}^{2} + b (- 3) + c$

Step 3

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

Solve for $a$ in $14 = a + b + c$ .

Rewrite the equation as $a + b + c = 14$ .

$a + b + c = 14$

$210 = a \cdot 5^{2} + b (5) + c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

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

Subtract $b$ from both sides of the equation.

$a + c = 14 - b$

$210 = a \cdot 5^{2} + b (5) + c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

Subtract $c$ from both sides of the equation.

$a = 14 - b - c$

$210 = a \cdot 5^{2} + b (5) + c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

$a = 14 - b - c$

$210 = a \cdot 5^{2} + b (5) + c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

$a = 14 - b - c$

$210 = a \cdot 5^{2} + b (5) + c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

Replace all occurrences of $a$ with $14 - b - c$ in each equation.

Replace all occurrences of $a$ in $210 = a \cdot 5^{2} + b (5) + c$ with $14 - b - c$ .

$210 = (14 - b - c) \cdot 5^{2} + b (5) + c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

Simplify the right side.

Simplify $(14 - b - c) \cdot 5^{2} + b (5) + c$ .

Simplify each term.

Raise $5$ to the power of $2$ .

$210 = (14 - b - c) \cdot 25 + b (5) + c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

Apply the distributive property.

$210 = 14 \cdot 25 - b \cdot 25 - c \cdot 25 + b (5) + c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

Simplify.

Multiply $14$ by $25$ .

$210 = 350 - b \cdot 25 - c \cdot 25 + b (5) + c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

Multiply $25$ by $- 1$ .

$210 = 350 - 25 b - c \cdot 25 + b (5) + c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

Multiply $25$ by $- 1$ .

$210 = 350 - 25 b - 25 c + b (5) + c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

$210 = 350 - 25 b - 25 c + b (5) + c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

Move $5$ to the left of $b$ .

$210 = 350 - 25 b - 25 c + 5 b + c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

$210 = 350 - 25 b - 25 c + 5 b + c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

Simplify by adding terms.

Add $- 25 b$ and $5 b$ .

$210 = 350 - 20 b - 25 c + c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

Add $- 25 c$ and $c$ .

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$10 = a {(- 3)}^{2} + b (- 3) + c$

Replace all occurrences of $a$ in $10 = a {(- 3)}^{2} + b (- 3) + c$ with $14 - b - c$ .

$10 = (14 - b - c) {(- 3)}^{2} + b (- 3) + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Simplify the right side.

Simplify $(14 - b - c) {(- 3)}^{2} + b (- 3) + c$ .

Simplify each term.

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

$10 = (14 - b - c) \cdot 9 + b (- 3) + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Apply the distributive property.

$10 = 14 \cdot 9 - b \cdot 9 - c \cdot 9 + b (- 3) + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Simplify.

Multiply $14$ by $9$ .

$10 = 126 - b \cdot 9 - c \cdot 9 + b (- 3) + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Multiply $9$ by $- 1$ .

$10 = 126 - 9 b - c \cdot 9 + b (- 3) + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Multiply $9$ by $- 1$ .

$10 = 126 - 9 b - 9 c + b (- 3) + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$10 = 126 - 9 b - 9 c + b (- 3) + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

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

$10 = 126 - 9 b - 9 c - 3 b + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$10 = 126 - 9 b - 9 c - 3 b + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Simplify by adding terms.

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

$10 = 126 - 12 b - 9 c + c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Add $- 9 c$ and $c$ .

$10 = 126 - 12 b - 8 c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$10 = 126 - 12 b - 8 c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$10 = 126 - 12 b - 8 c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$10 = 126 - 12 b - 8 c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$10 = 126 - 12 b - 8 c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Solve for $b$ in $10 = 126 - 12 b - 8 c$ .

Rewrite the equation as $126 - 12 b - 8 c = 10$ .

$126 - 12 b - 8 c = 10$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

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

Subtract $126$ from both sides of the equation.

$- 12 b - 8 c = 10 - 126$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

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

$- 12 b = 10 - 126 + 8 c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Subtract $126$ from $10$ .

$- 12 b = - 116 + 8 c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$- 12 b = - 116 + 8 c$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Divide each term in $- 12 b = - 116 + 8 c$ by $- 12$ and simplify.

Divide each term in $- 12 b = - 116 + 8 c$ by $- 12$ .

$\frac{- 12 b}{- 12} = \frac{- 116}{- 12} + \frac{8 c}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 b}{- 12} = \frac{- 116}{- 12} + \frac{8 c}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Divide $b$ by $1$ .

$b = \frac{- 116}{- 12} + \frac{8 c}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$b = \frac{- 116}{- 12} + \frac{8 c}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$b = \frac{- 116}{- 12} + \frac{8 c}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 116$ and $- 12$ .

Factor $- 4$ out of $- 116$ .

$b = \frac{- 4 \cdot 29}{- 12} + \frac{8 c}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Cancel the common factors.

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

$b = \frac{- 4 \cdot 29}{- 4 \cdot 3} + \frac{8 c}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Cancel the common factor.

$b = \frac{- 4 \cdot 29}{- 4 \cdot 3} + \frac{8 c}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Rewrite the expression.

$b = \frac{29}{3} + \frac{8 c}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$b = \frac{29}{3} + \frac{8 c}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$b = \frac{29}{3} + \frac{8 c}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

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

Factor $4$ out of $8 c$ .

$b = \frac{29}{3} + \frac{4 (2 c)}{- 12}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Cancel the common factors.

Factor $4$ out of $- 12$ .

$b = \frac{29}{3} + \frac{4 (2 c)}{4 (- 3)}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Cancel the common factor.

$b = \frac{29}{3} + \frac{4 (2 c)}{4 \cdot - 3}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Rewrite the expression.

$b = \frac{29}{3} + \frac{2 c}{- 3}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$b = \frac{29}{3} + \frac{2 c}{- 3}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$b = \frac{29}{3} + \frac{2 c}{- 3}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Move the negative in front of the fraction.

$b = \frac{29}{3} - \frac{2 c}{3}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$210 = 350 - 20 b - 24 c$

$a = 14 - b - c$

Replace all occurrences of $b$ with $\frac{29}{3} - \frac{2 c}{3}$ in each equation.

Replace all occurrences of $b$ in $210 = 350 - 20 b - 24 c$ with $\frac{29}{3} - \frac{2 c}{3}$ .

$210 = 350 - 20 (\frac{29}{3} - \frac{2 c}{3}) - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Simplify the right side.

Simplify $350 - 20 (\frac{29}{3} - \frac{2 c}{3}) - 24 c$ .

Simplify each term.

Apply the distributive property.

$210 = 350 - 20 (\frac{29}{3}) - 20 (- \frac{2 c}{3}) - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Multiply $- 20 (\frac{29}{3})$ .

Combine $- 20$ and $\frac{29}{3}$ .

$210 = 350 + \frac{- 20 \cdot 29}{3} - 20 (- \frac{2 c}{3}) - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Multiply $- 20$ by $29$ .

$210 = 350 + \frac{- 580}{3} - 20 (- \frac{2 c}{3}) - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

$210 = 350 + \frac{- 580}{3} - 20 (- \frac{2 c}{3}) - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Multiply $- 20 (- \frac{2 c}{3})$ .

Multiply $- 1$ by $- 20$ .

$210 = 350 + \frac{- 580}{3} + 20 (\frac{2 c}{3}) - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Combine $20$ and $\frac{2 c}{3}$ .

$210 = 350 + \frac{- 580}{3} + \frac{20 (2 c)}{3} - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Multiply $2$ by $20$ .

$210 = 350 + \frac{- 580}{3} + \frac{40 c}{3} - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

$210 = 350 + \frac{- 580}{3} + \frac{40 c}{3} - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Move the negative in front of the fraction.

$210 = 350 - \frac{580}{3} + \frac{40 c}{3} - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

$210 = 350 - \frac{580}{3} + \frac{40 c}{3} - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

To write $350$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$210 = 350 \cdot \frac{3}{3} - \frac{580}{3} + \frac{40 c}{3} - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Combine $350$ and $\frac{3}{3}$ .

$210 = \frac{350 \cdot 3}{3} - \frac{580}{3} + \frac{40 c}{3} - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Combine the numerators over the common denominator.

$210 = \frac{350 \cdot 3 - 580}{3} + \frac{40 c}{3} - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Simplify the numerator.

Multiply $350$ by $3$ .

$210 = \frac{1050 - 580}{3} + \frac{40 c}{3} - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Subtract $580$ from $1050$ .

$210 = \frac{470}{3} + \frac{40 c}{3} - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

$210 = \frac{470}{3} + \frac{40 c}{3} - 24 c$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

To write $- 24 c$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$210 = \frac{470}{3} + \frac{40 c}{3} - 24 c \cdot \frac{3}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Combine $- 24 c$ and $\frac{3}{3}$ .

$210 = \frac{470}{3} + \frac{40 c}{3} + \frac{- 24 c \cdot 3}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Combine the numerators over the common denominator.

$210 = \frac{470}{3} + \frac{40 c - 24 c \cdot 3}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Combine the numerators over the common denominator.

$210 = \frac{470 + 40 c - 24 c \cdot 3}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Multiply $3$ by $- 24$ .

$210 = \frac{470 + 40 c - 72 c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Subtract $72 c$ from $40 c$ .

$210 = \frac{470 - 32 c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Factor $2$ out of $470 - 32 c$ .

Factor $2$ out of $470$ .

$210 = \frac{2 (235) - 32 c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Factor $2$ out of $- 32 c$ .

$210 = \frac{2 (235) + 2 (- 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Factor $2$ out of $2 (235) + 2 (- 16 c)$ .

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - b - c$

Replace all occurrences of $b$ in $a = 14 - b - c$ with $\frac{29}{3} - \frac{2 c}{3}$ .

$a = 14 - (\frac{29}{3} - \frac{2 c}{3}) - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Simplify the right side.

Simplify $14 - (\frac{29}{3} - \frac{2 c}{3}) - c$ .

Simplify each term.

Apply the distributive property.

$a = 14 - \frac{29}{3} + \frac{2 c}{3} - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

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

Multiply $- 1$ by $- 1$ .

$a = 14 - \frac{29}{3} + 1 (\frac{2 c}{3}) - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

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

$a = 14 - \frac{29}{3} + \frac{2 c}{3} - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - \frac{29}{3} + \frac{2 c}{3} - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 14 - \frac{29}{3} + \frac{2 c}{3} - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

To write $14$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a = 14 \cdot \frac{3}{3} - \frac{29}{3} + \frac{2 c}{3} - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Combine $14$ and $\frac{3}{3}$ .

$a = \frac{14 \cdot 3}{3} - \frac{29}{3} + \frac{2 c}{3} - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{14 \cdot 3 - 29}{3} + \frac{2 c}{3} - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Simplify the numerator.

Multiply $14$ by $3$ .

$a = \frac{42 - 29}{3} + \frac{2 c}{3} - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Subtract $29$ from $42$ .

$a = \frac{13}{3} + \frac{2 c}{3} - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = \frac{13}{3} + \frac{2 c}{3} - c$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

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

$a = \frac{13}{3} + \frac{2 c}{3} - c \cdot \frac{3}{3}$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

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

$a = \frac{13}{3} + \frac{2 c}{3} + \frac{- c \cdot 3}{3}$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{13}{3} + \frac{2 c - c \cdot 3}{3}$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Combine the numerators over the common denominator.

$a = \frac{13 + 2 c - c \cdot 3}{3}$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Multiply $3$ by $- 1$ .

$a = \frac{13 + 2 c - 3 c}{3}$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Subtract $3 c$ from $2 c$ .

$a = \frac{13 - c}{3}$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = \frac{13 - c}{3}$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = \frac{13 - c}{3}$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = \frac{13 - c}{3}$

$210 = \frac{2 (235 - 16 c)}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Solve for $c$ in $210 = \frac{2 (235 - 16 c)}{3}$ .

Rewrite the equation as $\frac{2 (235 - 16 c)}{3} = 210$ .

$\frac{2 (235 - 16 c)}{3} = 210$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Multiply both sides by $3$ .

$\frac{2 (235 - 16 c)}{3} \cdot 3 = 210 \cdot 3$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Simplify.

Simplify the left side.

Simplify $\frac{2 (235 - 16 c)}{3} \cdot 3$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{2 (235 - 16 c)}{3} \cdot 3 = 210 \cdot 3$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Rewrite the expression.

$2 (235 - 16 c) = 210 \cdot 3$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$2 (235 - 16 c) = 210 \cdot 3$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Apply the distributive property.

$2 \cdot 235 + 2 (- 16 c) = 210 \cdot 3$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Simplify the expression.

Multiply $2$ by $235$ .

$470 + 2 (- 16 c) = 210 \cdot 3$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Multiply $- 16$ by $2$ .

$470 - 32 c = 210 \cdot 3$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Reorder $470$ and $- 32 c$ .

$- 32 c + 470 = 210 \cdot 3$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$- 32 c + 470 = 210 \cdot 3$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$- 32 c + 470 = 210 \cdot 3$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$- 32 c + 470 = 210 \cdot 3$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Simplify the right side.

Multiply $210$ by $3$ .

$- 32 c + 470 = 630$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$- 32 c + 470 = 630$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$- 32 c + 470 = 630$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Solve for $c$ .

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

Subtract $470$ from both sides of the equation.

$- 32 c = 630 - 470$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Subtract $470$ from $630$ .

$- 32 c = 160$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$- 32 c = 160$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Divide each term in $- 32 c = 160$ by $- 32$ and simplify.

Divide each term in $- 32 c = 160$ by $- 32$ .

$\frac{- 32 c}{- 32} = \frac{160}{- 32}$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Simplify the left side.

Cancel the common factor of $- 32$ .

Cancel the common factor.

$\frac{- 32 c}{- 32} = \frac{160}{- 32}$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Divide $c$ by $1$ .

$c = \frac{160}{- 32}$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$c = \frac{160}{- 32}$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$c = \frac{160}{- 32}$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

Simplify the right side.

Divide $160$ by $- 32$ .

$c = - 5$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$c = - 5$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$c = - 5$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$c = - 5$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

$c = - 5$

$a = \frac{13 - c}{3}$

$b = \frac{29}{3} - \frac{2 c}{3}$

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

Replace all occurrences of $c$ in $a = \frac{13 - c}{3}$ with $- 5$ .

$a = \frac{13 - (- 5)}{3}$

$c = - 5$

$b = \frac{29}{3} - \frac{2 c}{3}$

Simplify the right side.

Simplify $\frac{13 - (- 5)}{3}$ .

Simplify the numerator.

Multiply $- 1$ by $- 5$ .

$a = \frac{13 + 5}{3}$

$c = - 5$

$b = \frac{29}{3} - \frac{2 c}{3}$

Add $13$ and $5$ .

$a = \frac{18}{3}$

$c = - 5$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = \frac{18}{3}$

$c = - 5$

$b = \frac{29}{3} - \frac{2 c}{3}$

Divide $18$ by $3$ .

$a = 6$

$c = - 5$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 6$

$c = - 5$

$b = \frac{29}{3} - \frac{2 c}{3}$

$a = 6$

$c = - 5$

$b = \frac{29}{3} - \frac{2 c}{3}$

Replace all occurrences of $c$ in $b = \frac{29}{3} - \frac{2 c}{3}$ with $- 5$ .

$b = \frac{29}{3} - \frac{2 (- 5)}{3}$

$a = 6$

$c = - 5$

Simplify the right side.

Simplify $\frac{29}{3} - \frac{2 (- 5)}{3}$ .

Combine the numerators over the common denominator.

$b = \frac{29 - 2 \cdot - 5}{3}$

$a = 6$

$c = - 5$

Simplify the expression.

Multiply $- 2$ by $- 5$ .

$b = \frac{29 + 10}{3}$

$a = 6$

$c = - 5$

Add $29$ and $10$ .

$b = \frac{39}{3}$

$a = 6$

$c = - 5$

Divide $39$ by $3$ .

$b = 13$

$a = 6$

$c = - 5$

$b = 13$

$a = 6$

$c = - 5$

$b = 13$

$a = 6$

$c = - 5$

$b = 13$

$a = 6$

$c = - 5$

$b = 13$

$a = 6$

$c = - 5$

List all of the solutions.

$b = 13, a = 6, c = - 5$

Step 4

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

$y = 6 x^{2} + 13 x - 5$

Step 5

Find the Equation of the Parabola (5,210) , (1,14) , (-3,10)

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