Find the Equation of the Parabola (0,5) , (3,2) , (-1,10)

$(0, 5)$ , $(3, 2)$ , $(- 1, 10)$

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.

$5 = a {(0)}^{2} + b (0) + c$

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

$10 = a {(- 1)}^{2} + b (- 1) + c$

Step 3

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

Solve for $c$ in $5 = a \cdot 0^{2} + c$ .

Rewrite the equation as $a \cdot 0^{2} + c = 5$ .

$a \cdot 0^{2} + c = 5$

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

$10 = a {(- 1)}^{2} + b (- 1) + c$

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

Simplify each term.

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

$a \cdot 0 + c = 5$

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

$10 = a {(- 1)}^{2} + b (- 1) + c$

Multiply $a$ by $0$ .

$0 + c = 5$

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

$10 = a {(- 1)}^{2} + b (- 1) + c$

$0 + c = 5$

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

$10 = a {(- 1)}^{2} + b (- 1) + c$

Add $0$ and $c$ .

$c = 5$

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

$10 = a {(- 1)}^{2} + b (- 1) + c$

$c = 5$

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

$10 = a {(- 1)}^{2} + b (- 1) + c$

$c = 5$

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

$10 = a {(- 1)}^{2} + b (- 1) + c$

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

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

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

$c = 5$

$10 = a {(- 1)}^{2} + b (- 1) + c$

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

Simplify the left side.

Remove parentheses.

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

$c = 5$

$10 = a {(- 1)}^{2} + b (- 1) + c$

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

$c = 5$

$10 = a {(- 1)}^{2} + b (- 1) + c$

Simplify the right side.

Simplify each term.

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

$2 = a \cdot 9 + b (3) + 5$

$c = 5$

$10 = a {(- 1)}^{2} + b (- 1) + c$

Move $9$ to the left of $a$ .

$2 = 9 \cdot a + b (3) + 5$

$c = 5$

$10 = a {(- 1)}^{2} + b (- 1) + c$

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

$2 = 9 a + 3 b + 5$

$c = 5$

$10 = a {(- 1)}^{2} + b (- 1) + c$

$2 = 9 a + 3 b + 5$

$c = 5$

$10 = a {(- 1)}^{2} + b (- 1) + c$

$2 = 9 a + 3 b + 5$

$c = 5$

$10 = a {(- 1)}^{2} + b (- 1) + c$

$2 = 9 a + 3 b + 5$

$c = 5$

$10 = a {(- 1)}^{2} + b (- 1) + c$

Replace all occurrences of $c$ in $10 = a {(- 1)}^{2} + b (- 1) + c$ with $5$ .

$10 = a {(- 1)}^{2} + b (- 1) + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

Simplify $10 = a {(- 1)}^{2} + b (- 1) + 5$ .

Simplify the left side.

Remove parentheses.

$10 = a {(- 1)}^{2} + b (- 1) + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

$10 = a {(- 1)}^{2} + b (- 1) + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

Simplify the right side.

Simplify each term.

Raise $- 1$ to the power of $2$ .

$10 = a \cdot 1 + b (- 1) + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

Multiply $a$ by $1$ .

$10 = a + b (- 1) + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

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

$10 = a - 1 b + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

Rewrite $- 1 b$ as $- b$ .

$10 = a - b + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

$10 = a - b + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

$10 = a - b + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

$10 = a - b + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

$10 = a - b + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

Solve for $a$ in $10 = a - b + 5$ .

Rewrite the equation as $a - b + 5 = 10$ .

$a - b + 5 = 10$

$2 = 9 a + 3 b + 5$

$c = 5$

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

Add $b$ to both sides of the equation.

$a + 5 = 10 + b$

$2 = 9 a + 3 b + 5$

$c = 5$

Subtract $5$ from both sides of the equation.

$a = 10 + b - 5$

$2 = 9 a + 3 b + 5$

$c = 5$

Subtract $5$ from $10$ .

$a = b + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

$a = b + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

$a = b + 5$

$2 = 9 a + 3 b + 5$

$c = 5$

Replace all occurrences of $a$ with $b + 5$ in each equation.

Replace all occurrences of $a$ in $2 = 9 a + 3 b + 5$ with $b + 5$ .

$2 = 9 (b + 5) + 3 b + 5$

$a = b + 5$

$c = 5$

Simplify the right side.

Simplify $9 (b + 5) + 3 b + 5$ .

Simplify each term.

Apply the distributive property.

$2 = 9 b + 9 \cdot 5 + 3 b + 5$

$a = b + 5$

$c = 5$

Multiply $9$ by $5$ .

$2 = 9 b + 45 + 3 b + 5$

$a = b + 5$

$c = 5$

$2 = 9 b + 45 + 3 b + 5$

$a = b + 5$

$c = 5$

Simplify by adding terms.

Add $9 b$ and $3 b$ .

$2 = 12 b + 45 + 5$

$a = b + 5$

$c = 5$

Add $45$ and $5$ .

$2 = 12 b + 50$

$a = b + 5$

$c = 5$

$2 = 12 b + 50$

$a = b + 5$

$c = 5$

$2 = 12 b + 50$

$a = b + 5$

$c = 5$

$2 = 12 b + 50$

$a = b + 5$

$c = 5$

$2 = 12 b + 50$

$a = b + 5$

$c = 5$

Solve for $b$ in $2 = 12 b + 50$ .

Rewrite the equation as $12 b + 50 = 2$ .

$12 b + 50 = 2$

$a = b + 5$

$c = 5$

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

Subtract $50$ from both sides of the equation.

$12 b = 2 - 50$

$a = b + 5$

$c = 5$

Subtract $50$ from $2$ .

$12 b = - 48$

$a = b + 5$

$c = 5$

$12 b = - 48$

$a = b + 5$

$c = 5$

Divide each term in $12 b = - 48$ by $12$ and simplify.

Divide each term in $12 b = - 48$ by $12$ .

$\frac{12 b}{12} = \frac{- 48}{12}$

$a = b + 5$

$c = 5$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 b}{12} = \frac{- 48}{12}$

$a = b + 5$

$c = 5$

Divide $b$ by $1$ .

$b = \frac{- 48}{12}$

$a = b + 5$

$c = 5$

$b = \frac{- 48}{12}$

$a = b + 5$

$c = 5$

$b = \frac{- 48}{12}$

$a = b + 5$

$c = 5$

Simplify the right side.

Divide $- 48$ by $12$ .

$b = - 4$

$a = b + 5$

$c = 5$

$b = - 4$

$a = b + 5$

$c = 5$

$b = - 4$

$a = b + 5$

$c = 5$

$b = - 4$

$a = b + 5$

$c = 5$

Replace all occurrences of $b$ with $- 4$ in each equation.

Replace all occurrences of $b$ in $a = b + 5$ with $- 4$ .

$a = (- 4) + 5$

$b = - 4$

$c = 5$

Simplify the right side.

Add $- 4$ and $5$ .

$a = 1$

$b = - 4$

$c = 5$

$a = 1$

$b = - 4$

$c = 5$

$a = 1$

$b = - 4$

$c = 5$

List all of the solutions.

$a = 1, b = - 4, c = 5$

Step 4

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

$y = x^{2} - 4 x + 5$

Step 5

Find the Equation of the Parabola (0,5) , (3,2) , (-1,10)

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