They have common root

Algebra Level 3

If the equations, x 2 + a x + b = 0 x 2 + b x + a = 0 x^2 + ax + b = 0 \\ x^2 + bx + a = 0 have a common root α \alpha then which of the following options might be true?

Select one or more

a + b + 1 = 0 a+b+1=0 a + b = 1 a+b=1 α = 1 \alpha = 1 α + 1 = 0 \alpha +1=0

Ram Mohith by Ram Mohith , 18, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Tom Engelsman
Oct 20, 2018

Let us write the factored forms of each quadratic equation:

x 2 + a x + b = ( x − α ) ( x − b α ) = 0 x^2 + ax + b = (x - \alpha)(x - \frac{b}{\alpha}) = 0

x 2 + b x + a = ( x − α ) ( x − a α ) = 0 x^2 + bx + a = (x - \alpha)(x - \frac{a}{\alpha}) = 0

By Vieta's Formula, we now obtain the following coefficient relationships:

α + b α = − a \alpha + \frac{b}{\alpha} = -a (i)

α + a α = − b \alpha + \frac{a}{\alpha} = -b (ii)

Subtracting (ii) from (i) will produce b − a α = b − a ⇒ α = 1 \frac{b-a}{\alpha} = b -a \Rightarrow \boxed{\alpha = 1} . Substituting this value back into either (i) or (ii) produces a + b + 1 = 0 \boxed{a + b + 1 = 0} . Hence, choice 3 is correct.

2 Helpful 1 Interesting 0 Brilliant 0 Confused

1 pending report

Vote up reports you agree with

×

Problem Loading...

Note Loading...

Set Loading...