# Precalculus Examples

, ,

Step 1

Use the standard form of a quadratic equation as the starting point for finding the equation through the three points.

Step 2

Create a system of equations by substituting the and values of each point into the standard formula of a quadratic equation to create the three equation system.

Step 3

Step 3.1

Solve for in .

Step 3.1.1

Rewrite the equation as .

Step 3.1.2

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

Step 3.1.2.1

Subtract from both sides of the equation.

Step 3.1.2.2

Subtract from both sides of the equation.

Step 3.2

Replace all occurrences of with in each equation.

Step 3.2.1

Replace all occurrences of in with .

Step 3.2.2

Simplify the right side.

Step 3.2.2.1

Simplify .

Step 3.2.2.1.1

Simplify each term.

Step 3.2.2.1.1.1

Raise to the power of .

Step 3.2.2.1.1.2

Apply the distributive property.

Step 3.2.2.1.1.3

Multiply by .

Step 3.2.2.1.1.4

Multiply by .

Step 3.2.2.1.1.5

Move to the left of .

Step 3.2.2.1.2

Simplify by adding terms.

Step 3.2.2.1.2.1

Add and .

Step 3.2.2.1.2.2

Add and .

Step 3.2.3

Replace all occurrences of in with .

Step 3.2.4

Simplify the right side.

Step 3.2.4.1

Simplify .

Step 3.2.4.1.1

Simplify each term.

Step 3.2.4.1.1.1

Raise to the power of .

Step 3.2.4.1.1.2

Apply the distributive property.

Step 3.2.4.1.1.3

Multiply by .

Step 3.2.4.1.1.4

Multiply by .

Step 3.2.4.1.1.5

Move to the left of .

Step 3.2.4.1.2

Simplify by adding terms.

Step 3.2.4.1.2.1

Add and .

Step 3.2.4.1.2.2

Add and .

Step 3.3

Solve for in .

Step 3.3.1

Rewrite the equation as .

Step 3.3.2

Add to both sides of the equation.

Step 3.3.3

Divide each term in by and simplify.

Step 3.3.3.1

Divide each term in by .

Step 3.3.3.2

Simplify the left side.

Step 3.3.3.2.1

Cancel the common factor of .

Step 3.3.3.2.1.1

Cancel the common factor.

Step 3.3.3.2.1.2

Divide by .

Step 3.3.3.3

Simplify the right side.

Step 3.3.3.3.1

Simplify each term.

Step 3.3.3.3.1.1

Dividing two negative values results in a positive value.

Step 3.3.3.3.1.2

Cancel the common factor of and .

Step 3.3.3.3.1.2.1

Factor out of .

Step 3.3.3.3.1.2.2

Cancel the common factors.

Step 3.3.3.3.1.2.2.1

Factor out of .

Step 3.3.3.3.1.2.2.2

Cancel the common factor.

Step 3.3.3.3.1.2.2.3

Rewrite the expression.

Step 3.3.3.3.1.3

Move the negative in front of the fraction.

Step 3.4

Replace all occurrences of with in each equation.

Step 3.4.1

Replace all occurrences of in with .

Step 3.4.2

Simplify the right side.

Step 3.4.2.1

Simplify .

Step 3.4.2.1.1

Simplify each term.

Step 3.4.2.1.1.1

Apply the distributive property.

Step 3.4.2.1.1.2

Cancel the common factor of .

Step 3.4.2.1.1.2.1

Factor out of .

Step 3.4.2.1.1.2.2

Cancel the common factor.

Step 3.4.2.1.1.2.3

Rewrite the expression.

Step 3.4.2.1.1.3

Multiply by .

Step 3.4.2.1.1.4

Cancel the common factor of .

Step 3.4.2.1.1.4.1

Move the leading negative in into the numerator.

Step 3.4.2.1.1.4.2

Factor out of .

Step 3.4.2.1.1.4.3

Cancel the common factor.

Step 3.4.2.1.1.4.4

Rewrite the expression.

Step 3.4.2.1.1.5

Multiply by .

Step 3.4.2.1.2

Subtract from .

Step 3.4.3

Replace all occurrences of in with .

Step 3.4.4

Simplify the right side.

Step 3.4.4.1

Simplify .

Step 3.4.4.1.1

Simplify each term.

Step 3.4.4.1.1.1

Apply the distributive property.

Step 3.4.4.1.1.2

Multiply .

Step 3.4.4.1.1.2.1

Multiply by .

Step 3.4.4.1.1.2.2

Multiply by .

Step 3.4.4.1.2

To write as a fraction with a common denominator, multiply by .

Step 3.4.4.1.3

Simplify terms.

Step 3.4.4.1.3.1

Combine and .

Step 3.4.4.1.3.2

Combine the numerators over the common denominator.

Step 3.4.4.1.4

Simplify each term.

Step 3.4.4.1.4.1

Simplify the numerator.

Step 3.4.4.1.4.1.1

Factor out of .

Step 3.4.4.1.4.1.1.1

Factor out of .

Step 3.4.4.1.4.1.1.2

Factor out of .

Step 3.4.4.1.4.1.1.3

Factor out of .

Step 3.4.4.1.4.1.2

Multiply by .

Step 3.4.4.1.4.1.3

Subtract from .

Step 3.4.4.1.4.2

Multiply by .

Step 3.5

Solve for in .

Step 3.5.1

Rewrite the equation as .

Step 3.5.2

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

Step 3.5.2.1

Add to both sides of the equation.

Step 3.5.2.2

Add and .

Step 3.6

Replace all occurrences of with in each equation.

Step 3.6.1

Replace all occurrences of in with .

Step 3.6.2

Simplify the right side.

Step 3.6.2.1

Simplify .

Step 3.6.2.1.1

To write as a fraction with a common denominator, multiply by .

Step 3.6.2.1.2

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Step 3.6.2.1.2.1

Multiply by .

Step 3.6.2.1.2.2

Multiply by .

Step 3.6.2.1.3

Combine the numerators over the common denominator.

Step 3.6.2.1.4

Simplify the numerator.

Step 3.6.2.1.4.1

Multiply by .

Step 3.6.2.1.4.2

Add and .

Step 3.6.3

Replace all occurrences of in with .

Step 3.6.4

Simplify the right side.

Step 3.6.4.1

Simplify .

Step 3.6.4.1.1

Multiply by .

Step 3.6.4.1.2

To write as a fraction with a common denominator, multiply by .

Step 3.6.4.1.3

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Step 3.6.4.1.3.1

Multiply by .

Step 3.6.4.1.3.2

Multiply by .

Step 3.6.4.1.4

Combine the numerators over the common denominator.

Step 3.6.4.1.5

Simplify the numerator.

Step 3.6.4.1.5.1

Multiply by .

Step 3.6.4.1.5.2

Subtract from .

Step 3.6.4.1.6

Move the negative in front of the fraction.

Step 3.7

List all of the solutions.

Step 4

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

Step 5