Try this beautiful problem from Algebra based on Quadratic equation.
Algebra based on Quadratic equation PRMO Problem 9
Suppose a,b are integers and a + b is a root of \(x^2 +ax+b=0\).What is the maximum possible
values of \( b^2 \)?
- $49$
- $81$
- $64$
Key Concepts
Algebra
quadratic equation
Factorization
Check the Answer
But try the problem first…
Answer:$81$
PRMO-2018, Problem 9
Pre College Mathematics
Try with Hints
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 \( (b^2)_{max} = 81\)
Other useful links
- https://www.youtube.com/watch?v=_HYhADRACPs
- https://www.cheenta.com/largest-and-smallest-numbers-amc-8-2006-problem-22/