Question:

Write down the quadratic equation whose roots are \(5\) and \(6\).

Solution:

Method 1.

If a quadratic equation has two roots \(5\) and \(6\), then the equation must have two factors as \((x-5)\) and \((x-6\)). Other than these two roots, there will be a real factor, let’s say \(k\).
So multiplying all these factors will give us the quadratic equation.
       \(k(x-5)(x-6)\) 
\(\implies k(x^2-11x+30)\)
This is our quadratic equation.

Method 2.
In general, if we are provided two roots of any quadratic equation \(\alpha\) and \(\beta\), then the quadratic equation will be as 
\(k[x^2-(\alpha+\beta)x+\alpha\beta]\) where \(k\) be any real number.
According to the above quadratic equation, we will get the following quadratic equation.
        \(k[x^2-(5+6)x+5\cdot6]\)
\(\implies k(x^2-11x+30)\)

Final answer.
So our final answer will be
\(\boxed{\implies k(x^2-11x+30)}\)