Question

In solving a problem that reduces to a quadratic equation, one student makes a mistake in the constant term and obtains $8$ and $2$ for roots. Another student makes a mistake only in the coefficient of first-degree term and finds $-9$ and $-1$ for roots. The correct equation is

• A. $x^2-10x+9=0$
• B. $x^2+10+9=0$
• C. $x^2-10x+16=0$
• D. $x^2-8x-9=0$

Answer 1

The correct option is A $x^2-10x+9=0$

Let the actual quadratic equation be $x^2+bx+c=0$

First student makes a mistake in constant. Let's say he took the constant as $d$

Hence the quadratic equation for the first student becomes $x^2+bx+d=0$

The roots for this equation are $8$ and $2$

Sum of roots $=-b=10$
$\Rightarrow b=-10$

Product of the roots $=16=d$

$\Rightarrow d=16$

The second student makes a mistake in the coefficient of first degree term. Let's say he considered it to be $e$ instead of $b$

The equation becomes $x^2+ex+c=0$ with roots $-9,-1$

Sum of the roots $=-10=-e$

$\Rightarrow e=10$

Product of the roots $=9=c$

$\Rightarrow c=9$

Hence the actual quadratic equation becomes $x^2-10x+9=0$