If α and β are the roots of the equation x^2 – 22x + 105 = 0, what is

Question

If α and β are the roots of the equation  x^2 – 22x + 105 = 0 , what is the value of |α – β| ?

A. 8
B. 6
C. 4
D. 2
E. 1

GMATinsight · Answer

Sum of the roots,  (α + β) = \frac{-b}{a} = \frac{-(-22)}{1} = 22

Product of the roots,  (α * β) = \frac{c}{a} = \frac{105}{1} = 105

(α - β)^2 = (α + β)^2 - 4*(α*β)

(α - β)^2 = (22)^2 - 4*(105) = 484-420 = 64

i.e. (α - β) = 8

Answer: Option A

frroldao · Answer

Solving the equation gives  11\pm\sqrt{106}  which... is none of the answers...  Bunuel  - can you help here?

Thanks

Bunuel · Answer

Thank you both. Edited the question.

anjanita · Answer

The answer is A. Please find below my 2 cents

The sum of the roots (α+β)=−b/a=−(−22)/1=22
The Product of the roots α*β = c/a = 105/1 = 105

Now, (α-β)^2 = (α+β)^2 - 4*α*β = (22^2) - (4*105) = 484 - 420 = 64

Therefore |α-β| = 8

Archit3110 · Answer

given quadratic eqn  x^2 – 22x + 105 = 0 
can be written as ( x-15)(x-7)=0
x=15,7 are its roots
so |α – β| = l15-7l; 8
option A

Neghineh · Answer

Do you mind explaining how you got to this step:  (α - β)^2 = (α + β)^2 - 4*(α*β)

Abramo · Answer

Still couldn't get the point of how do you get Sum and product

Sum of the roots, (α+β)=−ba=−(−22)1=22(α+β)=−ba=−(−22)1=22

Product of the roots, (α∗β)=ca=151=15(α∗β)=ca=151=15

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