3.3: The Quadratic Formula Mention how important the quadratic formula is because we can’t always factor every quadratic equation. Show why 3x^2+2x-7 doesn’t work.

Review A quadratic equation is an equation that can be written in the form 𝑎 𝑥 2 +𝑏𝑥+𝑐=0, where a, b, and c are real numbers and 𝑎≠0.

𝒙= −(𝒃)± (𝒃) 𝟐 −𝟒(𝒂)(𝒄) 𝟐(𝒂) Important Properties Quadratic Formula: The solutions of 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 are given by: 𝒙= −(𝒃)± (𝒃) 𝟐 −𝟒(𝒂)(𝒄) 𝟐(𝒂) The Quadratic Formula can be used to solve any quadratic equation!!! This formula finds the x-intercepts/zeros/roots of a quadratic function

Example #1 2 𝑥 2 −𝑥=4 2 𝑥 2 −𝑥−4=0 𝒂=𝟐, 𝒃=−𝟏, 𝒄=−𝟒 𝑥= −(𝑏)± (𝑏) 2 −4(𝑎)(𝑐) 2(𝑎) Example #1 Remember, we need it to look like 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 in order to use the Quadratic Formula. Otherwise, IT DOESN’T WORK!!! What are the solutions? Use the Quadratic Formula. 2 𝑥 2 −𝑥=4 2 𝑥 2 −𝑥−4=0 𝒂=𝟐, 𝒃=−𝟏, 𝒄=−𝟒 𝑥= −(−𝟏)± (−𝟏) 2 −4(𝟐)(−𝟒) 2(𝟐) 𝒙= 1± 1+32 4 = 1± 33 4

On Your Own 𝒂=𝟔, 𝒃=−𝟓, 𝒄=−𝟒 6 𝑥 2 −5𝑥−4=0 𝑥= −(𝑏)± (𝑏) 2 −4(𝑎)(𝑐) 2(𝑎) On Your Own Remember, we need it to look like 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 in order to use the Quadratic Formula. Otherwise, IT DOESN’T WORK!!! What are the solutions? Use the Quadratic Formula. 6 𝑥 2 −5𝑥−4=0 𝒂=𝟔, 𝒃=−𝟓, 𝒄=−𝟒 𝑥= −(−𝟓)± (−𝟓) 2 −4(𝟔)(−𝟒) 2(6) 𝒙= 5± 25+96 12 = 5± 121 12 = 5±11 12 = 5+11 12 and 5−11 12 = 16 12 and − 6 12 = 𝟒 𝟑 and −𝟏 𝟐

Example #2: Check your answer by factoring!!! 𝑥 2 +6𝑥+9=0 𝑥= −(𝑏)± (𝑏) 2 −4(𝑎)(𝑐) 2(𝑎) Example #2: Remember, we need it to look like 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 in order to use the Quadratic Formula and notice that it’s already done for us! What are the solutions? Use the Quadratic Formula. 𝑥 2 +6𝑥+9=0 𝒂=𝟏, 𝒃=𝟔, 𝒄=𝟗 𝑥= −(𝟔)± (𝟔) 2 −4(𝟏)(𝟗) 2(𝟏) 𝑥 2 +6𝑥+9=0 𝑥+3 𝑥+3 =0 𝑥+3=0 means 𝑥=−3 𝒙= −6± 36−36 2 = −6± 0 2 =−𝟑 Check your answer by factoring!!!

Example #3 3 𝑥 2 −4𝑥=−10 𝒂=3, 𝒃=−4, 𝒄=𝟏𝟎 3 𝑥 2 −4𝑥+10=0 𝑥= −(𝑏)± (𝑏) 2 −4(𝑎)(𝑐) 2(𝑎) Example #3 Remember, we need it to look like 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 in order to use the Quadratic Formula. Otherwise, IT DOESN’T WORK!!! What are the solutions? Use the Quadratic Formula. 3 𝑥 2 −4𝑥=−10 3 𝑥 2 −4𝑥+10=0 𝒂=3, 𝒃=−4, 𝒄=𝟏𝟎 𝑥= −(−4)± (−4) 2 −4(𝟑)(𝟏𝟎) 2(𝟑) 𝒙= 4± 16−120 6 = 4± −104 6 = 4±𝑖 104 6 = 4±2𝑖 26 6 = 𝟐±𝒊 𝟐𝟔 𝟑

On Your Own 7 𝑥 2 +2𝑥+4=0 𝒂=7, 𝒃=𝟐, 𝒄=𝟒 𝑥= −(𝟐)± (𝟐) 2 −4(𝟕)(𝟒) 2(𝟕) 𝑥= −(𝑏)± (𝑏) 2 −4(𝑎)(𝑐) 2(𝑎) On Your Own Remember, we need it to look like 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 in order to use the Quadratic Formula. Otherwise, IT DOESN’T WORK!!! What are the solutions? Use the Quadratic Formula. 7 𝑥 2 +2𝑥+3=−1 7 𝑥 2 +2𝑥+4=0 𝒂=7, 𝒃=𝟐, 𝒄=𝟒 𝑥= −(𝟐)± (𝟐) 2 −4(𝟕)(𝟒) 2(𝟕) 𝒙= −2± 4−112 14 = −2± −108 14 = −2±𝑖 108 14 = −2±6𝑖 3 14 = −𝟏±𝟑𝒊 𝟑 𝟕

Definition The discriminant of a quadratic equation in the form 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 is the value of the expression 𝑏 2 −4𝑎𝑐. So it’s the value inside the square root  𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 The discriminant If 𝑏 2 −4𝑎𝑐<𝟎, then there are no real solutions to the quadratic equation If 𝑏 2 −4𝑎𝑐=𝟎, then the quadratic equation has only one real zero. If 𝑏 2 −4𝑎𝑐>𝟎, then the quadratic equation has two real solutions.

Examples of 0, 1, and 2 Solutions: Two real solutions 𝑥 2 =4 gives 𝑥=−2, 2 One real solution 𝑥 2 =0 gives 𝑥=0 No real solutions 𝑥 2 =−4

Example #4: Discriminant = 𝑏 2 −4𝑎𝑐= −𝟑 2 −4 −𝟐 𝟓 =9+40=49 What is the number of real solutions of −2 𝑥 2 −3𝑥+5=0? Hint: This means we have to know the value of the discriminant! -2 𝑥 2 −3𝑥+5=0 𝒂=−𝟐, 𝒃=−𝟑, 𝒄=𝟓 Discriminant = 𝑏 2 −4𝑎𝑐= −𝟑 2 −4 −𝟐 𝟓 =9+40=49 Since this number is positive (i.e., > 0), we have two real solutions!

Example #5: Discriminant = 𝑏 2 −4𝑎𝑐= −𝟑 2 −4 𝟐 𝟓 =9−40=−31 What is the number of real solutions of 2 𝑥 2 −3𝑥+5=0? Hint: This means we have to know the value of the discriminant! 2 𝑥 2 −3𝑥+5=0 𝒂=𝟐, 𝒃=−𝟑, 𝒄=𝟓 Discriminant = 𝑏 2 −4𝑎𝑐= −𝟑 2 −4 𝟐 𝟓 =9−40=−31 Since this number is negative (i.e., < 0), we have no real solutions!

Note We have now seen three ways to solve 𝑎 𝑥 2 +𝑏𝑥+𝑐=0: By factoring By completing the square Using the Quadratic Formula Order you should try first: Factoring Quadratic Formula Completing the Square