The Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation that is in the form ax2 + bx + c = 0 Let’s take a look at what this formula looks like. I’ll have to write that 138 times before I’ll be able to remember it.

Solving Equations with the Quadratic Formula Find the roots of the following equation. Now let’s use the quadratic formula. Let’s solve by factoring first. 3 3 1 -28 1 Hey, it’s Sam Ting.

Solving Unfactorable Equations Find the roots of the following equation. I can’t factor that! Find the roots of the same equation to the nearest hundredth. -3 -3 2 -1 2

Finding Imaginary Roots Find the roots of the following equation in simplest a + bi form. -2 -2 1 5 4 4 1 9 1 1 I think I understand. Irrational and Imaginary Roots always occur in conjugate pairs.

One More Example -6 -6 1 9 1 Hey, it’s Sam Ting.

Using the Discriminant The Discriminant is the expression under the radical sign in the quadratic formula. Let’s take another look at that formula. The Discriminant tells the nature of the roots. The Nature of the Roots can be Real, Rational, and Unequal I Think I’d like to see some examples. Real, Irrational, and Unequal Real, Rational, and Equal Imaginary

Discriminant Examples Perfect Square Zero Non Perfect Square Negative

Overview of the Discriminant Perfect Square Zero Intersects x-axis at 1 Distinct Point Intersects x-axis at 2 Distinct Points Non-Perfect Square Negative Intersects x-axis at 2 Non-Distinct Points Does Not Intersect x-axis

Sum and Product of the Roots Do you need to solve the equation to find the sum and product? Equation & Roots Sum Product

Formulas to Find the Sum and Product of the Roots Sum of the Roots = Product of the Roots = Sum Product Notice that they are all the same answers that we got before. Equation

Problems Using the Sum and Product of the Roots For the following quadratic equation, by what amount does the product of the roots exceed the sum of the roots? Product of the Roots = Sum of the Roots = The product of the roots exceeds the sum of the roots by 2. If one root of the following quadratic equation is –1, find (a) The other root and (b) the value of k. (a) The other root is 4 (b) The two roots are –1 and 4. The sum is 3 and the product is –4. That was easy

Forming an Equation from the Roots Form an equation whose roots are x = 2 and x = –5. The factors are: and Form a quadratic equation that has 5 + 2i as one of its roots. Since imaginary roots occur in conjugate pairs, the roots are: and Assume a = 1 Assume a = 1

Using Sum & Product of the Roots to Form an Equation Write a quadratic equation that has the given roots. Sum = Product = Sum = Product =