Slideshow 22, Mathematics Mr Richard Sasaki, Room 307 Vieta’s Formulae Slideshow 22, Mathematics Mr Richard Sasaki, Room 307

Objectives Understand how two solutions for a quadratic equation can be written as two simultaneous equations Understand how to use these equations Solve such equations to find solutions for 𝑥

Vieta Vieta was a French man. He knew how to factorise an equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 and considered this as 𝑥 2 + 𝑏 𝑎 𝑥+ 𝑐 𝑎 =0. When we factorise this we get… 𝑥− 𝑥 1 𝑥− 𝑥 2 =0 for some two solutions 𝑥 1 and 𝑥 2 . How do they relate to 𝑏 𝑎 and 𝑐 𝑎 ? 𝑏 𝑎 We know that − 𝑥 1 + − 𝑥 2 = and − 𝑥 1 ∙ − 𝑥 2 = . 𝑐 𝑎 𝑥 1 + 𝑥 2 =− 𝑏 𝑎 So… and . 𝑥 1 ∙ 𝑥 2 = 𝑐 𝑎

Vieta’s Formulae 𝑥 1 + 𝑥 2 =− 𝑏 𝑎 and . 𝑥 1 ∙ 𝑥 2 = 𝑐 𝑎 In the simple case 𝑥 2 +𝑏𝑥+𝑐=0, we get… 𝑥 1 + 𝑥 2 =−𝑏 𝑥 1 ∙ 𝑥 2 =𝑐 Example Solve 𝑥 2 −3𝑥−10=0 using Vieta’s Formulae. We have 𝑏=−3 and 𝑐=−10 so… 𝑥 1 + 𝑥 2 =3 𝑥 1 𝑥 2 =−10 Now let’s try to find the solutions for 𝑥 1 and 𝑥 2 by solving the simultaneous equations above!

Vieta’s Formulae ① 𝑥 1 + 𝑥 2 =3 ② 𝑥 1 𝑥 2 =−10 𝑥 1 𝑥 2 =−10 It doesn’t matter which I’ll substitute into which. I will substitute ② into ①. − 10 𝑥 2 ② 𝑥 1 𝑥 2 =−10 ⇒ 𝑥 1 = − 10 𝑥 2 + 𝑥 2 =3 ① 𝑥 1 + 𝑥 2 =3 ⇒ ⇒ −10+ 𝑥 2 2 =3 𝑥 2 ⇒ 𝑥 2 2 −3 𝑥 2 −10=0 I’m back where I started!!

Vieta’s Formulae Because the formulae are derived from the quadratic anyway, we can’t solve them like that. Vieta’s formulae can only be used to solve quadratics visually! 𝑥 1 + 𝑥 2 =3 𝑥 1 𝑥 2 =−10 What numbers add together to make 3 and multiply together to make −10? −2 and 5! (It’s very similar to factorisation.) So the solutions are 𝑥=−2 and 5.

Vieta’s Formulae Let’s try another! Example Solve 𝑥 2 +13𝑥+42=0 using Vieta’s Formulae. We have 𝑏= , 𝑐= , so… 13 42 𝑥 1 + 𝑥 2 =−𝑏 𝑥 1 + 𝑥 2 =−13 ⇒ 𝑥 1 𝑥 2 =𝑐 𝑥 1 𝑥 2 =42 What numbers add together to make −13 and multiply together to make 42? −6 and −7! So the solutions are 𝑥=−6 and −7.

Answers - Easy 3 2 −3 2 −1, −2 7 10 −7 10 −2, −5 2 −3 −2 −3 1, −3 −5 6 2, 3 6 9 −6 9 −3 6 −7 −6 −7 −7, 1 10 24 −10 24 −4, −6 −10 25 10 25 5 −2 −35 2 −35 7, −5

Answers - Hard 14 49 −14 49 −7 −11 −26 11 −26 −2, 13 −13 −14 13 −14 −1, 14 −18 81 18 81 9 −9 −9 ±3 −17 72 17 72 8, 9 19 70 −19 70 −14,−5 2 −2 −2, 0 −169 −169 ±13

Vieta’s Formulae If the numbers are easy, it is a similar process solving equations where 𝑎≠1. 𝑥 1 + 𝑥 2 =− 𝑏 𝑎 and . 𝑥 1 ∙ 𝑥 2 = 𝑐 𝑎 Example Solve 2𝑥 2 −2𝑥−4=0 using Vieta’s Formulae. 𝑎=2, 𝑏=−2, 𝑐=−4, 𝑠𝑜… 𝑥 1 + 𝑥 2 =− 𝑏 𝑎 𝑥 1 + 𝑥 2 =1 ⇒ 𝑥 1 𝑥 2 = 𝑐 𝑎 𝑥 1 𝑥 2 =−2 What numbers add together to make 1 and multiply together to make −2? −1 and 2! So the solutions are 𝑥=−1 and 2.

Vieta’s Formulae Let’s try a harder example. Example Solve 2𝑥 2 +7𝑥−4=0 using Vieta’s Formulae. 𝑎=2, 𝑏=7, 𝑐=−4, 𝑠𝑜… 𝑥 1 + 𝑥 2 =− 𝑏 𝑎 𝑥 1 + 𝑥 2 =− 7 2 ⇒ 𝑥 1 𝑥 2 = 𝑐 𝑎 𝑥 1 𝑥 2 =−2 What numbers add together to make − 7 2 and multiply together to make −2? 1 2 and −4! So the solutions are 𝑥= 1 2 and −4.

Answers – Easy / Hard 𝑥=4 𝑜𝑟 −1 𝑥=3 𝑜𝑟 −2 𝑥=3 𝑜𝑟 −1 𝑥=4 𝑜𝑟 −4 𝑥=3 𝑜𝑟 −4 𝑥= 3 2 𝑜𝑟 −2 𝑥=−1 𝑥=−3 𝑜𝑟 −4 𝑥= 5 2 𝑜𝑟 −3 𝑥=−8 𝑜𝑟 7 𝑥= 1 3 𝑜𝑟 −2 𝑥=5 𝑜𝑟 −5