Solving Quadratic Equations by the Quadratic Formula

What does it mean to “solve”? When you are asked to “solve” a quadratic, it is asking you to find the roots.

THE QUADRATIC FORMULA

What do we do with it? Step 1: plug 0 in for y, and get the 0 by itself. Step 2: Just identify a, b, and c then substitute into the formula.

Example 𝑦= 𝑥 2 +4𝑥+3 a=1. b=4. c=3 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 𝑥= −4± 4 2 −4∙1∙3 2∙1

Example 1 continued 𝑥= −4± 4 2 𝑥= −4+2 2 𝑎𝑛𝑑 𝑥= −4−2 2 x=-1 and x=-3

WHY USE THE QUADRATIC FORMULA? The quadratic formula allows you to solve ANY quadratic equation, even if you cannot factor it.

Definition Definition:Discriminant is the piece under the radical: b2 – 4ac

WHY IS THE DISCRIMINANT IMPORTANT? The discriminant tells you the number and types of answers (roots) you will get. The discriminant can be +, –, or 0 Since the discriminant is under a radical, think about what it means if you have a positive or negative number or 0 under the radical.

WHAT THE DISCRIMINANT TELLS YOU! Value of the Discriminant Nature of the Solutions Negative 2 imaginary solutions Zero 1 Real Solution Positive 2 Real Solutions

Refresh What is an imaginary number? When you have a square root of a negative number. Called i. −1 =𝑖

Example with imaginary answer: 𝑥 2 −2𝑥+5=0 Try solving on your own…

𝑥 2 −2𝑥+5=0 𝑥= −(−2)± (−2) 2 −4∙1∙5 2∙1 Plug in a,b and c 𝑥= 2± −16 2∙1 Simplify under the radical 𝑥= 2±𝑖 16 2 Factor out the -1, it becomes i. 𝑥= 2±𝑖∙4 2 Take square root 𝑥=1+2𝑖 𝑎𝑛𝑑 𝑥=1−2𝑖 Divide by 2.

You try! Start with 1 and 4, then go on to the others.

Write a quadratic in general from: and the x-intercepts are 5 and 2. What can we do?

Solution Write it factored form (because that is what information we have. 𝑦=(𝑥−5)(𝑥−2) Then foil to get it in general 𝑦= 𝑥 2 −7𝑥−10

Homework: This is a massive assignment! 7.4 Worksheet 1 a-c 2 all 3 a-f 4 a-d 5 a-c