Try this beautiful problem from Algebra based Quadratic equation.
Quadratic equation Problem – AMC-10A, 2002- Problem 12
Both roots of the quadratic equation \(x^2 – 63x + k = 0\) are prime numbers. The number of possible values of \(k\) is
- \(0\)
- \(1\)
- \(2\)
- \(4\)
- more than \(4\)
Key Concepts
Algebra
Quadratic equation
prime numbers
Check the Answer
But try the problem first…
Answer: \(1\)
AMC-10A (2002) Problem 12
Pre College Mathematics
Try with Hints
First hint
The given equation is \(x^2 – 63x + k = 0\). Say that the roots are primes…
Comparing the equation with \(ax^2 +bx+c=0\) we get \(a=1 , b=-63 , c=k\).. Let \(m_1\) & \(m_2 \) be the roots of the given equation…
using vieta’s Formula we may sat that…\(m_1 + m_2 =-(- 63)=63\) and \(m_1 m_2 = k\)
can you finish the problem……..
Second Hint
Now the roots are prime. Sum of the two roots are \(63\) and product is \(k\)
Therefore one root must be \(2\) ,otherwise the sum would be even number
can you finish the problem……..
Final Step
So other root will be \(63-2\)=\(61\). Therefore product must be \(m_1m_2=122\)
Hence the answer is \(1\)
Other useful links
- https://www.cheenta.com/probability-in-divisibility-amc-10a-2003-problem-15/
- https://www.youtube.com/watch?v=PIBuksVSNhE
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