Try this beautiful problem from Algebra based on Quadratic equation.
Suppose a,b are integers and a + b is a root of \(x^2 +ax+b=0\).What is the maximum possible
values of \( b^2 \)?
Algebra
quadratic equation
Factorization
But try the problem first...
Answer:$81$
PRMO-2018, Problem 9
Pre College Mathematics
First hint
(‘a+b”) is the root of the equation therefore (“a+b”) must satisfy the given equation
Can you now finish the problem ..........
Second Hint
Discriminant is a perfect square
can you finish the problem........
Final Step
Given that \(‘a+b”\) is the root of the equation therefore \(“a+b”\) must satisfy the given equation
Therefore the given equation becomes ……
\((a+b)^2 +a(a+b)+b=0\)
\(\Rightarrow a^2 +2ab+b^2+a^2+ab+b=0\)
\(\Rightarrow 2a^2 +3ab+b^2+b=0\)
Now since “a” is an integer,Discriminant is a perfect square
\(\Rightarrow 9b^2 -8(b^2+b)=m^2\) (for some \(m \in \mathbb Z)\)
\(\Rightarrow (b-4)^2 -16=m^2\)
\(\Rightarrow (b-4+m)(b-4-m)=16\)
Therefore the possible cases are \(b-4+m=\pm 8\), \(b-4-m=\pm 2\),\(b-4+m=b-4-m=\pm 4\)
i.e b-4=5,-5,4,-4
\(\Rightarrow b =9,-1,8,0\)
Therefore \( (b^2)_{max} = 81\)
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From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:
