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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