GMAT exam question about finding a quadratic equation given mistakes in solving it

Question

In solving a problem on quadratic equation, one student makes a mistake in the constant term of the equation and gets $-3$ and $-2$ for the roots. Another student makes a mistake in the coefficient of first degree term and finds $-1$ and $-2$ for the roots. What is the correct equation?

What I did:- Let a and b be the roots of the equation For the first mistake $a+b=5$( Sum of roots) $ab=6$

For the second mistake $a+b=-3$ $ab=3$ Answer of this question is $x^2+5x+2$ How can we get the answer of this question when we have been wrong values? I will be obliged if anyone of you can help me with this question

Answer 1

The original equation is $ax^2+bx+c=0$.

The first equation is $ax^2+bx+c'=0$. Then $-\dfrac ba =(-3)+(-2)=-5$ and so $\dfrac ba =5$.

The second equation is $ax^2+b'x+c=0$. Then $\dfrac ca=(-1)\cdot(-2)=2$.

So, the original equation is $ax^2+5ax+2a=0$ or, equivalently, $x^2+5x+2=0$, since $a\ne 0$.

Note that we cannot say what the original equation really was, because we don't know $a$, but we can find the roots.

Answer 2

Assuming that the coefficient of $x^2$ is $1$, the equation will be $x^2+5x+2=0$. In fact:

• from the first mistake, we get that the sum of the coefficient of $x$ is $5$;
• from the second mistake, we get that the constant term is $2$.