1. Slideshow 22, Mathematics Mr Richard Sasaki
Vieta’s Formulae
Slideshow 22, Mathematics
Mr Richard Sasaki

2. 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

3. 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

4. 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!

5. 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!!

6. 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.

7. Vieta’s Formulae Let’s try another! Example We have 𝑏= , 𝑐= , so… 13
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.

8. 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

9. 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

10. 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.

11. 𝑥=4 𝑜𝑟 −1
𝑥=3 𝑜𝑟 −1
𝑥=3 𝑜𝑟 −4
𝑥=−1
𝑥=−8 𝑜𝑟 7
𝑥=5 𝑜𝑟 −5