1.3 Quadratic Equations College Algebra: Equations and Inequalities

Quadratic Equations Quadratic equation: is an equation of the form: ax 2 + bx + c =0 Forms of the Quadratic Equation: – Standard Form: – Zero Form: – Vertex Form:

Zero-Factor Theorem If (x – r)(x –s) = 0 Then (x – r) = 0 (x – s) = 0 Where r and s are roots Solutions to equation are:

Zero-Factor Theorem – Example Solve the equation

Square Root Property If c > 0, the equation x 2 = c has two real roots:

Completing the Square This is another method used to solve quadratic equations The goal is to convert the standard form into the Vertex form This creates a perfect square trinomial

Completing the Square When a = 1 1. Group x terms on one side of the equation 2. Half the coefficient of x and then square it 3. Add the number found in 2 to both sides 4. Factor the perfect square trinomial and combine right side 5. Solve using the square root property

Completing the Square When a > 1 1. Group x terms on one side of the equation Divide both side by a 2. Half the coefficient of x and then square it 3. Add the number found in 2 to both sides 4. Factor the perfect square trinomial and combine right side 5. Solve using the square root property

The Quadratic Formula Solutions to ax 2 + bx + c =0

The Quadratic Formula – Example Solve a=5, b =-9, c=-2

The Discriminant The discriminant tells the nature of the roots of the quadratic equation

The Discriminant – Examples

Summary Quadratic Equations Zero-Factor Theorem Square Root Property Completing the Square The Quadratic Formula The DiscriminantDiscriminant