What constant should be added in the given polynomial x2-3x+3 so that it has at least two zeroes?
Answers
If your question is correct, there is no need to add any constant to get two roots. But, if you have mistaken in posting the question, you can find the range of the constant as follows:
Suppose, we need to add a constant 'a' to get real roots. Then, the given equation can be rewritten as, x^2 -3x+(a+3)=0 …..(1)
Now, for eq.(1) to have real & distinct roots its discriminant(D) must be positive.
i.e. D>0
=>(-3)^2 -4×1×(a+3)>0
=> 9–4a-12>0
=> -4a>3
=> a< -3/4
If a= -3/4, roots will be real & equal.
P.S.- Here I have consider real roots to get the range of 'a'. You can also find range of 'a' in case of complex roots using D<0.
x^2 - 3x + 3 = (x-1.5)^2 +0.75 >0 for all x, and so it has no real zeros. If we add any real number strictly less than (- 0.75), it will have exactly two zeros. Being a quadratic polynomial it cannot have more than two zeros ever as addition of a constant stills keeps it as a quadratic polynomial.