Math CentralQuandaries & Queries


Question from Awudu, a student:

Given the roots of the quadratic equation ax^2+5x+c=0 are 5 and -2/3,find the
values of a,b and c.


I am going to assume the equation be $a x^2 + b x + c = 0.$

If $5$ and $\large \frac{-2}{3}$ are the roots of a quadratic equation then you know that

\[p x^2 + q x + r = (x - 5)\left(x + \frac{2}{3}\right)\]

for some real numbers $p, q$ and $r.$

Expand the right side and equate the like coefficients. This will give you values of $p, q$ and $r$ with $p = 1$ and $q$ and $r$ fractions. I expect you want $a, b$ and $c$ to be integers. How do you modify the equation $p x^2 + q x + r = 0$ so that the coefficients are integers and the roots are still $5$ and $\large \frac{-2}{3} \normalsize ?$


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