Finite Math Examples

(- 3, - 9)

$(- 3, - 9)$ ,

(6, - 6)

$(6, - 6)$ ,

(- 3, 2)

$(- 3, 2)$

Step 1

Use the standard form of a quadratic equation

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

$y = a x^{2} + b x + c$ as the starting point for finding the equation through the three points.

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

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

Step 2

Create a system of equations by substituting the

x

$x$ and

y

$y$ values of each point into the standard formula of a quadratic equation to create the three equation system.

- 9 = a {(- 3)}^{2} + b (- 3) + c, - 6 = a {(6)}^{2} + b (6) + c, 2 = a {(- 3)}^{2} + b (- 3) + c

$- 9 = a {(- 3)}^{2} + b (- 3) + c, - 6 = a {(6)}^{2} + b (6) + c, 2 = a {(- 3)}^{2} + b (- 3) + c$

Step 3

Solve the system of equations to find the values of

a

$a$ ,

b

$b$ , and

c

$c$ .

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Step 3.1

Solve for

c

$c$ in

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

$- 9 = a {(- 3)}^{2} + b (- 3) + c$ .

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Step 3.1.1

Rewrite the equation as

a {(- 3)}^{2} + b (- 3) + c = - 9

$a {(- 3)}^{2} + b (- 3) + c = - 9$ .

a {(- 3)}^{2} + b (- 3) + c = - 9

$a {(- 3)}^{2} + b (- 3) + c = - 9$

- 6 = a \cdot 6^{2} + b (6) + c

$- 6 = a \cdot 6^{2} + b (6) + c$

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

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

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Raise

- 3

$- 3$ to the power of

2

$2$ .

a \cdot 9 + b (- 3) + c = - 9

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

- 6 = a \cdot 6^{2} + b (6) + c

$- 6 = a \cdot 6^{2} + b (6) + c$

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

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

Step 3.1.2.2

Move

9

$9$ to the left of

a

$a$ .

9 \cdot a + b (- 3) + c = - 9

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

- 6 = a \cdot 6^{2} + b (6) + c

$- 6 = a \cdot 6^{2} + b (6) + c$

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

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

Step 3.1.2.3

Move

- 3

$- 3$ to the left of

b

$b$ .

9 a - 3 b + c = - 9

$9 a - 3 b + c = - 9$

- 6 = a \cdot 6^{2} + b (6) + c

$- 6 = a \cdot 6^{2} + b (6) + c$

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

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

9 a - 3 b + c = - 9

$9 a - 3 b + c = - 9$

- 6 = a \cdot 6^{2} + b (6) + c

$- 6 = a \cdot 6^{2} + b (6) + c$

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

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

Step 3.1.3

Move all terms not containing

c

$c$ to the right side of the equation.

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Step 3.1.3.1

Subtract

9 a

$9 a$ from both sides of the equation.

- 3 b + c = - 9 - 9 a

$- 3 b + c = - 9 - 9 a$

- 6 = a \cdot 6^{2} + b (6) + c

$- 6 = a \cdot 6^{2} + b (6) + c$

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

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

Step 3.1.3.2

Add

3 b

$3 b$ to both sides of the equation.

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

- 6 = a \cdot 6^{2} + b (6) + c

$- 6 = a \cdot 6^{2} + b (6) + c$

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

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

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

- 6 = a \cdot 6^{2} + b (6) + c

$- 6 = a \cdot 6^{2} + b (6) + c$

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

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

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

- 6 = a \cdot 6^{2} + b (6) + c

$- 6 = a \cdot 6^{2} + b (6) + c$

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

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

Step 3.2

Replace all occurrences of

c

$c$ with

- 9 - 9 a + 3 b

$- 9 - 9 a + 3 b$ in each equation.

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Step 3.2.1

Replace all occurrences of

c

$c$ in

- 6 = a \cdot 6^{2} + b (6) + c

$- 6 = a \cdot 6^{2} + b (6) + c$ with

- 9 - 9 a + 3 b

$- 9 - 9 a + 3 b$ .

- 6 = a \cdot 6^{2} + b (6) - 9 - 9 a + 3 b

$- 6 = a \cdot 6^{2} + b (6) - 9 - 9 a + 3 b$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

Step 3.2.2

Simplify

- 6 = a \cdot 6^{2} + b (6) - 9 - 9 a + 3 b

$- 6 = a \cdot 6^{2} + b (6) - 9 - 9 a + 3 b$ .

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Remove parentheses.

- 6 = a \cdot 6^{2} + b (6) - 9 - 9 a + 3 b

$- 6 = a \cdot 6^{2} + b (6) - 9 - 9 a + 3 b$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

- 6 = a \cdot 6^{2} + b (6) - 9 - 9 a + 3 b

$- 6 = a \cdot 6^{2} + b (6) - 9 - 9 a + 3 b$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Simplify

a \cdot 6^{2} + b (6) - 9 - 9 a + 3 b

$a \cdot 6^{2} + b (6) - 9 - 9 a + 3 b$ .

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Step 3.2.2.2.1.1

Simplify each term.

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Step 3.2.2.2.1.1.1

Raise

6

$6$ to the power of

2

$2$ .

- 6 = a \cdot 36 + b (6) - 9 - 9 a + 3 b

$- 6 = a \cdot 36 + b (6) - 9 - 9 a + 3 b$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

Step 3.2.2.2.1.1.2

Move

36

$36$ to the left of

a

$a$ .

- 6 = 36 \cdot a + b (6) - 9 - 9 a + 3 b

$- 6 = 36 \cdot a + b (6) - 9 - 9 a + 3 b$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

Step 3.2.2.2.1.1.3

Move

6

$6$ to the left of

b

$b$ .

- 6 = 36 a + 6 b - 9 - 9 a + 3 b

$- 6 = 36 a + 6 b - 9 - 9 a + 3 b$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

- 6 = 36 a + 6 b - 9 - 9 a + 3 b

$- 6 = 36 a + 6 b - 9 - 9 a + 3 b$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

Step 3.2.2.2.1.2

Simplify by adding terms.

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Step 3.2.2.2.1.2.1

Subtract

9 a

$9 a$ from

36 a

$36 a$ .

- 6 = 27 a + 6 b - 9 + 3 b

$- 6 = 27 a + 6 b - 9 + 3 b$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

Step 3.2.2.2.1.2.2

Add

6 b

$6 b$ and

3 b

$3 b$ .

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

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

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

Step 3.2.3

Replace all occurrences of

c

$c$ in

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

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

- 9 - 9 a + 3 b

$- 9 - 9 a + 3 b$ .

2 = a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b

$2 = a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

Step 3.2.4

Simplify

2 = a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b

$2 = a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b$ .

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Step 3.2.4.1

Simplify the left side.

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Step 3.2.4.1.1

Remove parentheses.

2 = a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b

$2 = a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

2 = a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b

$2 = a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

Step 3.2.4.2

Simplify the right side.

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Step 3.2.4.2.1

Simplify

a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b

$a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b$ .

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Step 3.2.4.2.1.1

Combine the opposite terms in

a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b

$a {(- 3)}^{2} + b (- 3) - 9 - 9 a + 3 b$ .

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Step 3.2.4.2.1.1.1

Reorder the factors in the terms

b (- 3)

$b (- 3)$ and

3 b

$3 b$ .

2 = a {(- 3)}^{2} - 3 b - 9 - 9 a + 3 b

$2 = a {(- 3)}^{2} - 3 b - 9 - 9 a + 3 b$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

Step 3.2.4.2.1.1.2

Add

- 3 b

$- 3 b$ and

3 b

$3 b$ .

2 = a {(- 3)}^{2} - 9 - 9 a + 0

$2 = a {(- 3)}^{2} - 9 - 9 a + 0$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

Step 3.2.4.2.1.1.3

Add

a {(- 3)}^{2} - 9 - 9 a

$a {(- 3)}^{2} - 9 - 9 a$ and

0

$0$ .

2 = a {(- 3)}^{2} - 9 - 9 a

$2 = a {(- 3)}^{2} - 9 - 9 a$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

2 = a {(- 3)}^{2} - 9 - 9 a

$2 = a {(- 3)}^{2} - 9 - 9 a$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

Step 3.2.4.2.1.2

Simplify each term.

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Step 3.2.4.2.1.2.1

Raise

- 3

$- 3$ to the power of

2

$2$ .

2 = a \cdot 9 - 9 - 9 a

$2 = a \cdot 9 - 9 - 9 a$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

Step 3.2.4.2.1.2.2

Move

9

$9$ to the left of

a

$a$ .

2 = 9 a - 9 - 9 a

$2 = 9 a - 9 - 9 a$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

2 = 9 a - 9 - 9 a

$2 = 9 a - 9 - 9 a$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

Step 3.2.4.2.1.3

Combine the opposite terms in

9 a - 9 - 9 a

$9 a - 9 - 9 a$ .

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Step 3.2.4.2.1.3.1

Subtract

9 a

$9 a$ from

9 a

$9 a$ .

2 = 0 - 9

$2 = 0 - 9$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

Step 3.2.4.2.1.3.2

Subtract

9

$9$ from

0

$0$ .

2 = - 9

$2 = - 9$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

2 = - 9

$2 = - 9$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

2 = - 9

$2 = - 9$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

2 = - 9

$2 = - 9$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

2 = - 9

$2 = - 9$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

2 = - 9

$2 = - 9$

- 6 = 27 a + 9 b - 9

$- 6 = 27 a + 9 b - 9$

c = - 9 - 9 a + 3 b

$c = - 9 - 9 a + 3 b$

Step 3.3

Since

2 = - 9

$2 = - 9$ is not true, there is no solution.

No solution