Question

If $a,b,c$ are odd integers and $a{x}^2+bx+c=0$ has real roots then:

• A. Both roots are positive
• B. Roots are of opposite signs.
• C. Both roots are rational
• D. Both roots are irrational

Answer 1

The correct option is D

Both roots are irrational

$a,b,c$ are odd integers, $D=b^2-4ac\ge 0$

If equation has rational roots, then $b^2-4ac$ is perfect square. Since $b^2-4ac$ (odd - even) is odd, $b^2-4ac$ is square of odd integer.

Let $b^2-4ac=(2k+1{)}^2$

$\Rightarrow 4ac=(2n+1{)}^2-(2k+1{)}^2$ $[b=2n+1]$

$=(2n+1-2k-1)(2n+2k+2)$

$=4(n-k)(n+k+1)$

$\Rightarrow (n-k)(n+k+1)=ac$

$\Rightarrow$ even interger = odd integer which is not true.

Hence, both roots must be irrational.