Find the Equation of the Parabola (0,-2) , (5,3) , (25,-77)

$(0, - 2)$ , $(5, 3)$ , $(25, - 77)$

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.

$- 2 = a {(0)}^{2} + b (0) + c$

$3 = a {(5)}^{2} + b (5) + c$

$- 77 = a {(25)}^{2} + b (25) + c$

Step 3

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

Solve for $c$ in $- 2 = a \cdot 0^{2} + c$ .

Rewrite the equation as $a \cdot 0^{2} + c = - 2$ .

$a \cdot 0^{2} + c = - 2$

$3 = a \cdot 5^{2} + b (5) + c$

$- 77 = a \cdot 25^{2} + b (25) + c$

Simplify $a \cdot 0^{2} + c$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$a \cdot 0 + c = - 2$

$3 = a \cdot 5^{2} + b (5) + c$

$- 77 = a \cdot 25^{2} + b (25) + c$

Multiply $a$ by $0$ .

$0 + c = - 2$

$3 = a \cdot 5^{2} + b (5) + c$

$- 77 = a \cdot 25^{2} + b (25) + c$

$0 + c = - 2$

$3 = a \cdot 5^{2} + b (5) + c$

$- 77 = a \cdot 25^{2} + b (25) + c$

Add $0$ and $c$ .

$c = - 2$

$3 = a \cdot 5^{2} + b (5) + c$

$- 77 = a \cdot 25^{2} + b (25) + c$

$c = - 2$

$3 = a \cdot 5^{2} + b (5) + c$

$- 77 = a \cdot 25^{2} + b (25) + c$

$c = - 2$

$3 = a \cdot 5^{2} + b (5) + c$

$- 77 = a \cdot 25^{2} + b (25) + c$

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

Replace all occurrences of $c$ in $3 = a \cdot 5^{2} + b (5) + c$ with $- 2$ .

$3 = a \cdot 5^{2} + b (5) - 2$

$c = - 2$

$- 77 = a \cdot 25^{2} + b (25) + c$

Simplify $3 = a \cdot 5^{2} + b (5) - 2$ .

Simplify the left side.

Remove parentheses.

$3 = a \cdot 5^{2} + b (5) - 2$

$c = - 2$

$- 77 = a \cdot 25^{2} + b (25) + c$

$3 = a \cdot 5^{2} + b (5) - 2$

$c = - 2$

$- 77 = a \cdot 25^{2} + b (25) + c$

Simplify the right side.

Simplify each term.

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

$3 = a \cdot 25 + b (5) - 2$

$c = - 2$

$- 77 = a \cdot 25^{2} + b (25) + c$

Move $25$ to the left of $a$ .

$3 = 25 \cdot a + b (5) - 2$

$c = - 2$

$- 77 = a \cdot 25^{2} + b (25) + c$

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

$3 = 25 a + 5 b - 2$

$c = - 2$

$- 77 = a \cdot 25^{2} + b (25) + c$

$3 = 25 a + 5 b - 2$

$c = - 2$

$- 77 = a \cdot 25^{2} + b (25) + c$

$3 = 25 a + 5 b - 2$

$c = - 2$

$- 77 = a \cdot 25^{2} + b (25) + c$

$3 = 25 a + 5 b - 2$

$c = - 2$

$- 77 = a \cdot 25^{2} + b (25) + c$

Replace all occurrences of $c$ in $- 77 = a \cdot 25^{2} + b (25) + c$ with $- 2$ .

$- 77 = a \cdot 25^{2} + b (25) - 2$

$3 = 25 a + 5 b - 2$

$c = - 2$

Simplify $- 77 = a \cdot 25^{2} + b (25) - 2$ .

Simplify the left side.

Remove parentheses.

$- 77 = a \cdot 25^{2} + b (25) - 2$

$3 = 25 a + 5 b - 2$

$c = - 2$

$- 77 = a \cdot 25^{2} + b (25) - 2$

$3 = 25 a + 5 b - 2$

$c = - 2$

Simplify the right side.

Simplify each term.

Raise $25$ to the power of $2$ .

$- 77 = a \cdot 625 + b (25) - 2$

$3 = 25 a + 5 b - 2$

$c = - 2$

Move $625$ to the left of $a$ .

$- 77 = 625 \cdot a + b (25) - 2$

$3 = 25 a + 5 b - 2$

$c = - 2$

Move $25$ to the left of $b$ .

$- 77 = 625 a + 25 b - 2$

$3 = 25 a + 5 b - 2$

$c = - 2$

$- 77 = 625 a + 25 b - 2$

$3 = 25 a + 5 b - 2$

$c = - 2$

$- 77 = 625 a + 25 b - 2$

$3 = 25 a + 5 b - 2$

$c = - 2$

$- 77 = 625 a + 25 b - 2$

$3 = 25 a + 5 b - 2$

$c = - 2$

$- 77 = 625 a + 25 b - 2$

$3 = 25 a + 5 b - 2$

$c = - 2$

Solve for $a$ in $- 77 = 625 a + 25 b - 2$ .

Rewrite the equation as $625 a + 25 b - 2 = - 77$ .

$625 a + 25 b - 2 = - 77$

$3 = 25 a + 5 b - 2$

$c = - 2$

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

Subtract $25 b$ from both sides of the equation.

$625 a - 2 = - 77 - 25 b$

$3 = 25 a + 5 b - 2$

$c = - 2$

Add $2$ to both sides of the equation.

$625 a = - 77 - 25 b + 2$

$3 = 25 a + 5 b - 2$

$c = - 2$

Add $- 77$ and $2$ .

$625 a = - 25 b - 75$

$3 = 25 a + 5 b - 2$

$c = - 2$

$625 a = - 25 b - 75$

$3 = 25 a + 5 b - 2$

$c = - 2$

Divide each term in $625 a = - 25 b - 75$ by $625$ and simplify.

Divide each term in $625 a = - 25 b - 75$ by $625$ .

$\frac{625 a}{625} = \frac{- 25 b}{625} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Simplify the left side.

Cancel the common factor of $625$ .

Cancel the common factor.

$\frac{625 a}{625} = \frac{- 25 b}{625} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Divide $a$ by $1$ .

$a = \frac{- 25 b}{625} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

$a = \frac{- 25 b}{625} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

$a = \frac{- 25 b}{625} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 25$ and $625$ .

Factor $25$ out of $- 25 b$ .

$a = \frac{25 (- b)}{625} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Cancel the common factors.

Factor $25$ out of $625$ .

$a = \frac{25 (- b)}{25 (25)} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Cancel the common factor.

$a = \frac{25 (- b)}{25 \cdot 25} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Rewrite the expression.

$a = \frac{- b}{25} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

$a = \frac{- b}{25} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

$a = \frac{- b}{25} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Move the negative in front of the fraction.

$a = - \frac{b}{25} + \frac{- 75}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Cancel the common factor of $- 75$ and $625$ .

Factor $25$ out of $- 75$ .

$a = - \frac{b}{25} + \frac{25 (- 3)}{625}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Cancel the common factors.

Factor $25$ out of $625$ .

$a = - \frac{b}{25} + \frac{25 \cdot - 3}{25 \cdot 25}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Cancel the common factor.

$a = - \frac{b}{25} + \frac{25 \cdot - 3}{25 \cdot 25}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Rewrite the expression.

$a = - \frac{b}{25} + \frac{- 3}{25}$

$3 = 25 a + 5 b - 2$

$c = - 2$

$a = - \frac{b}{25} + \frac{- 3}{25}$

$3 = 25 a + 5 b - 2$

$c = - 2$

$a = - \frac{b}{25} + \frac{- 3}{25}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Move the negative in front of the fraction.

$a = - \frac{b}{25} - \frac{3}{25}$

$3 = 25 a + 5 b - 2$

$c = - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$3 = 25 a + 5 b - 2$

$c = - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$3 = 25 a + 5 b - 2$

$c = - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$3 = 25 a + 5 b - 2$

$c = - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$3 = 25 a + 5 b - 2$

$c = - 2$

Replace all occurrences of $a$ with $- \frac{b}{25} - \frac{3}{25}$ in each equation.

Replace all occurrences of $a$ in $3 = 25 a + 5 b - 2$ with $- \frac{b}{25} - \frac{3}{25}$ .

$3 = 25 (- \frac{b}{25} - \frac{3}{25}) + 5 b - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Simplify the right side.

Simplify $25 (- \frac{b}{25} - \frac{3}{25}) + 5 b - 2$ .

Simplify each term.

Apply the distributive property.

$3 = 25 (- \frac{b}{25}) + 25 (- \frac{3}{25}) + 5 b - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Cancel the common factor of $25$ .

Move the leading negative in $- \frac{b}{25}$ into the numerator.

$3 = 25 (\frac{- b}{25}) + 25 (- \frac{3}{25}) + 5 b - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Cancel the common factor.

$3 = 25 (\frac{- b}{25}) + 25 (- \frac{3}{25}) + 5 b - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Rewrite the expression.

$3 = - b + 25 (- \frac{3}{25}) + 5 b - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$3 = - b + 25 (- \frac{3}{25}) + 5 b - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Cancel the common factor of $25$ .

Move the leading negative in $- \frac{3}{25}$ into the numerator.

$3 = - b + 25 (\frac{- 3}{25}) + 5 b - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Cancel the common factor.

$3 = - b + 25 (\frac{- 3}{25}) + 5 b - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Rewrite the expression.

$3 = - b - 3 + 5 b - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$3 = - b - 3 + 5 b - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$3 = - b - 3 + 5 b - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Simplify by adding terms.

Add $- b$ and $5 b$ .

$3 = 4 b - 3 - 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Subtract $2$ from $- 3$ .

$3 = 4 b - 5$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$3 = 4 b - 5$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$3 = 4 b - 5$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$3 = 4 b - 5$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$3 = 4 b - 5$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Solve for $b$ in $3 = 4 b - 5$ .

Rewrite the equation as $4 b - 5 = 3$ .

$4 b - 5 = 3$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

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

Add $5$ to both sides of the equation.

$4 b = 3 + 5$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Add $3$ and $5$ .

$4 b = 8$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$4 b = 8$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Divide each term in $4 b = 8$ by $4$ and simplify.

Divide each term in $4 b = 8$ by $4$ .

$\frac{4 b}{4} = \frac{8}{4}$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 b}{4} = \frac{8}{4}$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Divide $b$ by $1$ .

$b = \frac{8}{4}$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$b = \frac{8}{4}$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$b = \frac{8}{4}$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

Simplify the right side.

Divide $8$ by $4$ .

$b = 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$b = 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$b = 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

$b = 2$

$a = - \frac{b}{25} - \frac{3}{25}$

$c = - 2$

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

Replace all occurrences of $b$ in $a = - \frac{b}{25} - \frac{3}{25}$ with $2$ .

$a = - \frac{2}{25} - \frac{3}{25}$

$b = 2$

$c = - 2$

Simplify the right side.

Simplify $- \frac{2}{25} - \frac{3}{25}$ .

Combine the numerators over the common denominator.

$a = \frac{- 2 - 3}{25}$

$b = 2$

$c = - 2$

Subtract $3$ from $- 2$ .

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

$b = 2$

$c = - 2$

Cancel the common factor of $- 5$ and $25$ .

Factor $5$ out of $- 5$ .

$a = \frac{5 (- 1)}{25}$

$b = 2$

$c = - 2$

Cancel the common factors.

Factor $5$ out of $25$ .

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

$b = 2$

$c = - 2$

Cancel the common factor.

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

$b = 2$

$c = - 2$

Rewrite the expression.

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

$b = 2$

$c = - 2$

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

$b = 2$

$c = - 2$

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

$b = 2$

$c = - 2$

Move the negative in front of the fraction.

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

$b = 2$

$c = - 2$

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

$b = 2$

$c = - 2$

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

$b = 2$

$c = - 2$

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

$b = 2$

$c = - 2$

List all of the solutions.

$a = - \frac{1}{5}, b = 2, c = - 2$

Step 4

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

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

Step 5

Find the Equation of the Parabola (0,-2) , (5,3) , (25,-77)

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