Algebra Examples

$(0, 1)$ , $(2, 5)$ , $(3, 7)$

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.

$1 = a {(0)}^{2} + b (0) + c, 5 = a {(2)}^{2} + b (2) + c, 7 = a {(3)}^{2} + b (3) + c$

Step 3

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

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Step 3.1

Solve for $c$ in $1 = a \cdot 0^{2} + c$ .

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Step 3.1.1

Rewrite the equation as $a \cdot 0^{2} + c = 1$ .

$a \cdot 0^{2} + c = 1$

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

$7 = a \cdot 3^{2} + b (3) + c$

Step 3.1.2

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

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

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

$a \cdot 0 + c = 1$

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

$7 = a \cdot 3^{2} + b (3) + c$

Step 3.1.2.1.2

Multiply $a$ by $0$ .

$0 + c = 1$

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

$7 = a \cdot 3^{2} + b (3) + c$

$0 + c = 1$

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

$7 = a \cdot 3^{2} + b (3) + c$

Step 3.1.2.2

Add $0$ and $c$ .

$c = 1$

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

$7 = a \cdot 3^{2} + b (3) + c$

$c = 1$

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

$7 = a \cdot 3^{2} + b (3) + c$

$c = 1$

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

$7 = a \cdot 3^{2} + b (3) + c$

Step 3.2

Replace all occurrences of $c$ with $1$ in each equation.

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Step 3.2.1

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

$5 = a \cdot 2^{2} + b (2) + 1$

$c = 1$

$7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.2

Simplify $5 = a \cdot 2^{2} + b (2) + 1$ .

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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.

$5 = a \cdot 2^{2} + b (2) + 1$

$c = 1$

$7 = a \cdot 3^{2} + b (3) + c$

$5 = a \cdot 2^{2} + b (2) + 1$

$c = 1$

$7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Simplify each term.

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Step 3.2.2.2.1.1

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

$5 = a \cdot 4 + b (2) + 1$

$c = 1$

$7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.2.2.1.2

Move $4$ to the left of $a$ .

$5 = 4 \cdot a + b (2) + 1$

$c = 1$

$7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.2.2.1.3

Move $2$ to the left of $b$ .

$5 = 4 a + 2 b + 1$

$c = 1$

$7 = a \cdot 3^{2} + b (3) + c$

$5 = 4 a + 2 b + 1$

$c = 1$

$7 = a \cdot 3^{2} + b (3) + c$

$5 = 4 a + 2 b + 1$

$c = 1$

$7 = a \cdot 3^{2} + b (3) + c$

$5 = 4 a + 2 b + 1$

$c = 1$

$7 = a \cdot 3^{2} + b (3) + c$

Step 3.2.3

Replace all occurrences of $c$ in $7 = a \cdot 3^{2} + b (3) + c$ with $1$ .

$7 = a \cdot 3^{2} + b (3) + 1$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.2.4

Simplify $7 = a \cdot 3^{2} + b (3) + 1$ .

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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.

$7 = a \cdot 3^{2} + b (3) + 1$

$5 = 4 a + 2 b + 1$

$c = 1$

$7 = a \cdot 3^{2} + b (3) + 1$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.2.4.2

Simplify the right side.

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Step 3.2.4.2.1

Simplify each term.

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Step 3.2.4.2.1.1

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

$7 = a \cdot 9 + b (3) + 1$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.2.4.2.1.2

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

$7 = 9 \cdot a + b (3) + 1$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.2.4.2.1.3

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

$7 = 9 a + 3 b + 1$

$5 = 4 a + 2 b + 1$

$c = 1$

$7 = 9 a + 3 b + 1$

$5 = 4 a + 2 b + 1$

$c = 1$

$7 = 9 a + 3 b + 1$

$5 = 4 a + 2 b + 1$

$c = 1$

$7 = 9 a + 3 b + 1$

$5 = 4 a + 2 b + 1$

$c = 1$

$7 = 9 a + 3 b + 1$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3

Solve for $a$ in $7 = 9 a + 3 b + 1$ .

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Step 3.3.1

Rewrite the equation as $9 a + 3 b + 1 = 7$ .

$9 a + 3 b + 1 = 7$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $3 b$ from both sides of the equation.

$9 a + 1 = 7 - 3 b$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.2.2

Subtract $1$ from both sides of the equation.

$9 a = 7 - 3 b - 1$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.2.3

Subtract $1$ from $7$ .

$9 a = - 3 b + 6$

$5 = 4 a + 2 b + 1$

$c = 1$

$9 a = - 3 b + 6$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3

Divide each term in $9 a = - 3 b + 6$ by $9$ and simplify.

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Step 3.3.3.1

Divide each term in $9 a = - 3 b + 6$ by $9$ .

$\frac{9 a}{9} = \frac{- 3 b}{9} + \frac{6}{9}$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $9$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{9 a}{9} = \frac{- 3 b}{9} + \frac{6}{9}$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 3 b}{9} + \frac{6}{9}$

$5 = 4 a + 2 b + 1$

$c = 1$

$a = \frac{- 3 b}{9} + \frac{6}{9}$

$5 = 4 a + 2 b + 1$

$c = 1$

$a = \frac{- 3 b}{9} + \frac{6}{9}$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Simplify each term.

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Step 3.3.3.3.1.1

Cancel the common factor of $- 3$ and $9$ .

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Step 3.3.3.3.1.1.1

Factor $3$ out of $- 3 b$ .

$a = \frac{3 (- b)}{9} + \frac{6}{9}$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3.3.1.1.2

Cancel the common factors.

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Step 3.3.3.3.1.1.2.1

Factor $3$ out of $9$ .

$a = \frac{3 (- b)}{3 (3)} + \frac{6}{9}$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3.3.1.1.2.2

Cancel the common factor.

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

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{3} + \frac{6}{9}$

$5 = 4 a + 2 b + 1$

$c = 1$

$a = \frac{- b}{3} + \frac{6}{9}$

$5 = 4 a + 2 b + 1$

$c = 1$

$a = \frac{- b}{3} + \frac{6}{9}$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{3} + \frac{6}{9}$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3.3.1.3

Cancel the common factor of $6$ and $9$ .

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Step 3.3.3.3.1.3.1

Factor $3$ out of $6$ .

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

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3.3.1.3.2

Cancel the common factors.

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Step 3.3.3.3.1.3.2.1

Factor $3$ out of $9$ .

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

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3.3.1.3.2.2

Cancel the common factor.

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

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.3.3.3.1.3.2.3

Rewrite the expression.

$a = - \frac{b}{3} + \frac{2}{3}$

$5 = 4 a + 2 b + 1$

$c = 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$5 = 4 a + 2 b + 1$

$c = 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$5 = 4 a + 2 b + 1$

$c = 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$5 = 4 a + 2 b + 1$

$c = 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$5 = 4 a + 2 b + 1$

$c = 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$5 = 4 a + 2 b + 1$

$c = 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$5 = 4 a + 2 b + 1$

$c = 1$

Step 3.4

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

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Step 3.4.1

Replace all occurrences of $a$ in $5 = 4 a + 2 b + 1$ with $- \frac{b}{3} + \frac{2}{3}$ .

$5 = 4 (- \frac{b}{3} + \frac{2}{3}) + 2 b + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $4 (- \frac{b}{3} + \frac{2}{3}) + 2 b + 1$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Apply the distributive property.

$5 = 4 (- \frac{b}{3}) + 4 (\frac{2}{3}) + 2 b + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.1.2

Multiply $4 (- \frac{b}{3})$ .

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Step 3.4.2.1.1.2.1

Multiply $- 1$ by $4$ .

$5 = - 4 \frac{b}{3} + 4 (\frac{2}{3}) + 2 b + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.1.2.2

Combine $- 4$ and $\frac{b}{3}$ .

$5 = \frac{- 4 b}{3} + 4 (\frac{2}{3}) + 2 b + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$5 = \frac{- 4 b}{3} + 4 (\frac{2}{3}) + 2 b + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.1.3

Multiply $4 (\frac{2}{3})$ .

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Step 3.4.2.1.1.3.1

Combine $4$ and $\frac{2}{3}$ .

$5 = \frac{- 4 b}{3} + \frac{4 \cdot 2}{3} + 2 b + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.1.3.2

Multiply $4$ by $2$ .

$5 = \frac{- 4 b}{3} + \frac{8}{3} + 2 b + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$5 = \frac{- 4 b}{3} + \frac{8}{3} + 2 b + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.1.4

Move the negative in front of the fraction.

$5 = - \frac{4 b}{3} + \frac{8}{3} + 2 b + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$5 = - \frac{4 b}{3} + \frac{8}{3} + 2 b + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.2

To write $2 b$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$5 = - \frac{4 b}{3} + 2 b \cdot \frac{3}{3} + \frac{8}{3} + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.3

Simplify terms.

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Step 3.4.2.1.3.1

Combine $2 b$ and $\frac{3}{3}$ .

$5 = - \frac{4 b}{3} + \frac{2 b \cdot 3}{3} + \frac{8}{3} + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.3.2

Combine the numerators over the common denominator.

$5 = \frac{- 4 b + 2 b \cdot 3}{3} + \frac{8}{3} + 1$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.3.3

Combine the numerators over the common denominator.

$5 = 1 + \frac{- 4 b + 2 b \cdot 3 + 8}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.3.4

Multiply $3$ by $2$ .

$5 = 1 + \frac{- 4 b + 6 b + 8}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.3.5

Add $- 4 b$ and $6 b$ .

$5 = 1 + \frac{2 b + 8}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.3.6

Factor $2$ out of $2 b + 8$ .

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Step 3.4.2.1.3.6.1

Factor $2$ out of $2 b$ .

$5 = 1 + \frac{2 (b) + 8}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.3.6.2

Factor $2$ out of $8$ .

$5 = 1 + \frac{2 b + 2 \cdot 4}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.3.6.3

Factor $2$ out of $2 b + 2 \cdot 4$ .

$5 = 1 + \frac{2 (b + 4)}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$5 = 1 + \frac{2 (b + 4)}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.3.7

Write $1$ as a fraction with a common denominator.

$5 = \frac{3}{3} + \frac{2 (b + 4)}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.3.8

Combine the numerators over the common denominator.

$5 = \frac{3 + 2 (b + 4)}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$5 = \frac{3 + 2 (b + 4)}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.4

Simplify the numerator.

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Step 3.4.2.1.4.1

Apply the distributive property.

$5 = \frac{3 + 2 b + 2 \cdot 4}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.4.2

Multiply $2$ by $4$ .

$5 = \frac{3 + 2 b + 8}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.4.2.1.4.3

Add $3$ and $8$ .

$5 = \frac{2 b + 11}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$5 = \frac{2 b + 11}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$5 = \frac{2 b + 11}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$5 = \frac{2 b + 11}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$5 = \frac{2 b + 11}{3}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.5

Solve for $b$ in $5 = \frac{2 b + 11}{3}$ .

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Step 3.5.1

Rewrite the equation as $\frac{2 b + 11}{3} = 5$ .

$\frac{2 b + 11}{3} = 5$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.5.2

Multiply both sides by $3$ .

$\frac{2 b + 11}{3} \cdot 3 = 5 \cdot 3$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.5.3

Simplify.

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Step 3.5.3.1

Simplify the left side.

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Step 3.5.3.1.1

Cancel the common factor of $3$ .

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Step 3.5.3.1.1.1

Cancel the common factor.

$\frac{2 b + 11}{3} \cdot 3 = 5 \cdot 3$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.5.3.1.1.2

Rewrite the expression.

$2 b + 11 = 5 \cdot 3$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$2 b + 11 = 5 \cdot 3$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$2 b + 11 = 5 \cdot 3$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.5.3.2

Simplify the right side.

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Step 3.5.3.2.1

Multiply $5$ by $3$ .

$2 b + 11 = 15$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$2 b + 11 = 15$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$2 b + 11 = 15$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.5.4

Solve for $b$ .

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Step 3.5.4.1

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

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Step 3.5.4.1.1

Subtract $11$ from both sides of the equation.

$2 b = 15 - 11$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.5.4.1.2

Subtract $11$ from $15$ .

$2 b = 4$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$2 b = 4$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.5.4.2

Divide each term in $2 b = 4$ by $2$ and simplify.

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Step 3.5.4.2.1

Divide each term in $2 b = 4$ by $2$ .

$\frac{2 b}{2} = \frac{4}{2}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.5.4.2.2

Simplify the left side.

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Step 3.5.4.2.2.1

Cancel the common factor of $2$ .

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Step 3.5.4.2.2.1.1

Cancel the common factor.

$\frac{2 b}{2} = \frac{4}{2}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.5.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{4}{2}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$b = \frac{4}{2}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$b = \frac{4}{2}$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.5.4.2.3

Simplify the right side.

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Step 3.5.4.2.3.1

Divide $4$ by $2$ .

$b = 2$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$b = 2$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$b = 2$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$b = 2$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

$b = 2$

$a = - \frac{b}{3} + \frac{2}{3}$

$c = 1$

Step 3.6

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

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Step 3.6.1

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

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

$b = 2$

$c = 1$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

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

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Step 3.6.2.1.1

Combine the numerators over the common denominator.

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

$b = 2$

$c = 1$

Step 3.6.2.1.2

Simplify the expression.

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Step 3.6.2.1.2.1

Add $- 2$ and $2$ .

$a = \frac{0}{3}$

$b = 2$

$c = 1$

Step 3.6.2.1.2.2

Divide $0$ by $3$ .

$a = 0$

$b = 2$

$c = 1$

$a = 0$

$b = 2$

$c = 1$

$a = 0$

$b = 2$

$c = 1$

$a = 0$

$b = 2$

$c = 1$

$a = 0$

$b = 2$

$c = 1$

Step 3.7

List all of the solutions.

$a = 0, b = 2, c = 1$

Step 4

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

$y = 2 x + 1$

Step 5