Question:
What is the quadratic equation if the 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.
\(k[x^2-(5+6)x+5\cdot6]\)
\(\implies k(x^2-11x+30)\)
So our final answer will be
\(\boxed{\implies k(x^2-11x+30)}\)
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