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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

- \(0\)
- \(1\)
- \(2\)
- \(4\)
- more than \(4\)

Algebra

Quadratic equation

prime numbers

But try the problem first...

Answer: \(1\)

Source

Suggested Reading

AMC-10A (2002) Problem 12

Pre College Mathematics

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 \(1\)

- https://www.cheenta.com/probability-in-divisibility-amc-10a-2003-problem-15/
- https://www.youtube.com/watch?v=PIBuksVSNhE

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