# Quadratic equation Problem | AMC-10A, 2002 | Problem 12

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

prime numbers

Answer: $1$

AMC-10A (2002) Problem 12

Pre College Mathematics

## Try with Hints

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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