Try this beautiful problem from Algebra based Quadratic equation.
Both roots of the quadratic equation \(x^2 - 63x + k = 0\) are prime numbers. The number of possible values of \(k\) is
Algebra
Quadratic equation
prime numbers
Answer: \(1\)
AMC-10A (2002) Problem 12
Pre College Mathematics
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........
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........
So other root will be \(63-2\)=\(61\). Therefore product must be \(m_1m_2=122\)
Hence the answer is \(1\)
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