Finite Math Examples

$(0, 5)$ , $(- 1, 5.25)$ , $(5, 18.75)$

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.

$5 = a {(0)}^{2} + b (0) + c, 5.25 = a {(- 1)}^{2} + b (- 1) + c, 18.75 = a {(5)}^{2} + b (5) + c$

Step 3

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

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

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

$5.25 = a {(- 1)}^{2} + b (- 1) + c$

$18.75 = a \cdot 5^{2} + b (5) + c$

Step 3.1.2

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

Tap for more steps...

Step 3.1.2.1

Simplify each term.

Tap for more steps...

Step 3.1.2.1.1

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

$a \cdot 0 + c = 5$

$5.25 = a {(- 1)}^{2} + b (- 1) + c$

$18.75 = a \cdot 5^{2} + b (5) + c$

Step 3.1.2.1.2

Multiply $a$ by $0$ .

$0 + c = 5$

$5.25 = a {(- 1)}^{2} + b (- 1) + c$

$18.75 = a \cdot 5^{2} + b (5) + c$

$0 + c = 5$

$5.25 = a {(- 1)}^{2} + b (- 1) + c$

$18.75 = a \cdot 5^{2} + b (5) + c$

Step 3.1.2.2

Add $0$ and $c$ .

$c = 5$

$5.25 = a {(- 1)}^{2} + b (- 1) + c$

$18.75 = a \cdot 5^{2} + b (5) + c$

$c = 5$

$5.25 = a {(- 1)}^{2} + b (- 1) + c$

$18.75 = a \cdot 5^{2} + b (5) + c$

$c = 5$

$5.25 = a {(- 1)}^{2} + b (- 1) + c$

$18.75 = a \cdot 5^{2} + b (5) + c$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Replace all occurrences of $c$ in $5.25 = a {(- 1)}^{2} + b (- 1) + c$ with $5$ .

$5.25 = a {(- 1)}^{2} + b (- 1) + 5$

$c = 5$

$18.75 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2

Simplify $5.25 = a {(- 1)}^{2} + b (- 1) + 5$ .

Tap for more steps...

Step 3.2.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.2.1.1

Remove parentheses.

$5.25 = a {(- 1)}^{2} + b (- 1) + 5$

$c = 5$

$18.75 = a \cdot 5^{2} + b (5) + c$

$5.25 = a {(- 1)}^{2} + b (- 1) + 5$

$c = 5$

$18.75 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.2.1.1

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

$5.25 = a \cdot 1 + b (- 1) + 5$

$c = 5$

$18.75 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2.2.1.2

Multiply $a$ by $1$ .

$5.25 = a + b (- 1) + 5$

$c = 5$

$18.75 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2.2.1.3

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

$5.25 = a - 1 b + 5$

$c = 5$

$18.75 = a \cdot 5^{2} + b (5) + c$

Step 3.2.2.2.1.4

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

$5.25 = a - b + 5$

$c = 5$

$18.75 = a \cdot 5^{2} + b (5) + c$

$5.25 = a - b + 5$

$c = 5$

$18.75 = a \cdot 5^{2} + b (5) + c$

$5.25 = a - b + 5$

$c = 5$

$18.75 = a \cdot 5^{2} + b (5) + c$

$5.25 = a - b + 5$

$c = 5$

$18.75 = a \cdot 5^{2} + b (5) + c$

Step 3.2.3

Replace all occurrences of $c$ in $18.75 = a \cdot 5^{2} + b (5) + c$ with $5$ .

$18.75 = a \cdot 5^{2} + b (5) + 5$

$5.25 = a - b + 5$

$c = 5$

Step 3.2.4

Simplify $18.75 = a \cdot 5^{2} + b (5) + 5$ .

Tap for more steps...

Step 3.2.4.1

Simplify the left side.

Tap for more steps...

Step 3.2.4.1.1

Remove parentheses.

$18.75 = a \cdot 5^{2} + b (5) + 5$

$5.25 = a - b + 5$

$c = 5$

$18.75 = a \cdot 5^{2} + b (5) + 5$

$5.25 = a - b + 5$

$c = 5$

Step 3.2.4.2

Simplify the right side.

Tap for more steps...

Step 3.2.4.2.1

Simplify each term.

Tap for more steps...

Step 3.2.4.2.1.1

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

$18.75 = a \cdot 25 + b (5) + 5$

$5.25 = a - b + 5$

$c = 5$

Step 3.2.4.2.1.2

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

$18.75 = 25 \cdot a + b (5) + 5$

$5.25 = a - b + 5$

$c = 5$

Step 3.2.4.2.1.3

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

$18.75 = 25 a + 5 b + 5$

$5.25 = a - b + 5$

$c = 5$

$18.75 = 25 a + 5 b + 5$

$5.25 = a - b + 5$

$c = 5$

$18.75 = 25 a + 5 b + 5$

$5.25 = a - b + 5$

$c = 5$

$18.75 = 25 a + 5 b + 5$

$5.25 = a - b + 5$

$c = 5$

$18.75 = 25 a + 5 b + 5$

$5.25 = a - b + 5$

$c = 5$

Step 3.3

Solve for $a$ in $5.25 = a - b + 5$ .

Tap for more steps...

Step 3.3.1

Rewrite the equation as $a - b + 5 = 5.25$ .

$a - b + 5 = 5.25$

$18.75 = 25 a + 5 b + 5$

$c = 5$

Step 3.3.2

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

Tap for more steps...

Step 3.3.2.1

Add $b$ to both sides of the equation.

$a + 5 = 5.25 + b$

$18.75 = 25 a + 5 b + 5$

$c = 5$

Step 3.3.2.2

Subtract $5$ from both sides of the equation.

$a = 5.25 + b - 5$

$18.75 = 25 a + 5 b + 5$

$c = 5$

Step 3.3.2.3

Subtract $5$ from $5.25$ .

$a = b + 0.25$

$18.75 = 25 a + 5 b + 5$

$c = 5$

$a = b + 0.25$

$18.75 = 25 a + 5 b + 5$

$c = 5$

$a = b + 0.25$

$18.75 = 25 a + 5 b + 5$

$c = 5$

Step 3.4

Replace all occurrences of $a$ with $b + 0.25$ in each equation.

Tap for more steps...

Step 3.4.1

Replace all occurrences of $a$ in $18.75 = 25 a + 5 b + 5$ with $b + 0.25$ .

$18.75 = 25 (b + 0.25) + 5 b + 5$

$a = b + 0.25$

$c = 5$

Step 3.4.2

Simplify the right side.

Tap for more steps...

Step 3.4.2.1

Simplify $25 (b + 0.25) + 5 b + 5$ .

Tap for more steps...

Step 3.4.2.1.1

Simplify each term.

Tap for more steps...

Step 3.4.2.1.1.1

Apply the distributive property.

$18.75 = 25 b + 25 \cdot 0.25 + 5 b + 5$

$a = b + 0.25$

$c = 5$

Step 3.4.2.1.1.2

Multiply $25$ by $0.25$ .

$18.75 = 25 b + 6.25 + 5 b + 5$

$a = b + 0.25$

$c = 5$

$18.75 = 25 b + 6.25 + 5 b + 5$

$a = b + 0.25$

$c = 5$

Step 3.4.2.1.2

Simplify by adding terms.

Tap for more steps...

Step 3.4.2.1.2.1

Add $25 b$ and $5 b$ .

$18.75 = 30 b + 6.25 + 5$

$a = b + 0.25$

$c = 5$

Step 3.4.2.1.2.2

Add $6.25$ and $5$ .

$18.75 = 30 b + 11.25$

$a = b + 0.25$

$c = 5$

$18.75 = 30 b + 11.25$

$a = b + 0.25$

$c = 5$

$18.75 = 30 b + 11.25$

$a = b + 0.25$

$c = 5$

$18.75 = 30 b + 11.25$

$a = b + 0.25$

$c = 5$

$18.75 = 30 b + 11.25$

$a = b + 0.25$

$c = 5$

Step 3.5

Solve for $b$ in $18.75 = 30 b + 11.25$ .

Tap for more steps...

Step 3.5.1

Rewrite the equation as $30 b + 11.25 = 18.75$ .

$30 b + 11.25 = 18.75$

$a = b + 0.25$

$c = 5$

Step 3.5.2

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

Tap for more steps...

Step 3.5.2.1

Subtract $11.25$ from both sides of the equation.

$30 b = 18.75 - 11.25$

$a = b + 0.25$

$c = 5$

Step 3.5.2.2

Subtract $11.25$ from $18.75$ .

$30 b = 7.5$

$a = b + 0.25$

$c = 5$

$30 b = 7.5$

$a = b + 0.25$

$c = 5$

Step 3.5.3

Divide each term in $30 b = 7.5$ by $30$ and simplify.

Tap for more steps...

Step 3.5.3.1

Divide each term in $30 b = 7.5$ by $30$ .

$\frac{30 b}{30} = \frac{7.5}{30}$

$a = b + 0.25$

$c = 5$

Step 3.5.3.2

Simplify the left side.

Tap for more steps...

Step 3.5.3.2.1

Cancel the common factor of $30$ .

Tap for more steps...

Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{30 b}{30} = \frac{7.5}{30}$

$a = b + 0.25$

$c = 5$

Step 3.5.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{7.5}{30}$

$a = b + 0.25$

$c = 5$

$b = \frac{7.5}{30}$

$a = b + 0.25$

$c = 5$

$b = \frac{7.5}{30}$

$a = b + 0.25$

$c = 5$

Step 3.5.3.3

Simplify the right side.

Tap for more steps...

Step 3.5.3.3.1

Divide $7.5$ by $30$ .

$b = 0.25$

$a = b + 0.25$

$c = 5$

$b = 0.25$

$a = b + 0.25$

$c = 5$

$b = 0.25$

$a = b + 0.25$

$c = 5$

$b = 0.25$

$a = b + 0.25$

$c = 5$

Step 3.6

Replace all occurrences of $b$ with $0.25$ in each equation.

Tap for more steps...

Step 3.6.1

Replace all occurrences of $b$ in $a = b + 0.25$ with $0.25$ .

$a = (0.25) + 0.25$

$b = 0.25$

$c = 5$

Step 3.6.2

Simplify the right side.

Tap for more steps...

Step 3.6.2.1

Add $0.25$ and $0.25$ .

$a = 0.5$

$b = 0.25$

$c = 5$

$a = 0.5$

$b = 0.25$

$c = 5$

$a = 0.5$

$b = 0.25$

$c = 5$

Step 3.7

List all of the solutions.

$a = 0.5, b = 0.25, c = 5$

Step 4

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

$y = 0.5 x^{2} + 0.25 x + 5$

Step 5