Question

The equation x^2 + ax + b = 0 has solutions x=a and x=b where a and b are different.
How many equations are there?

A 0 B 1 C 3 D 4 E an infinity

Answer 1

if x=a and x=b are solutions, then we know the equation can be written as:\[(x-a)(x-b)=0\]so, if we expand the left-hand-side here, we get:\[x^2-ax-bx+ab=0\]\[x^2-(a+b)x+ab=0\]this has to match our original equation which was:\[x^2+ax+b=0\]so we can equate the coefficients of each power of x to get the following equations:\[a=-(a+b)=-a-b\]\[b=ab\]

the b's in the 2nd equation cancel to give us a=1. putting this value of a into the 1st of these two equations gives us:\[1=-1-b\]so b=-2.

this means we have one unique solution, so answer is B.