Question

The number of the integral values of b, which the equation \[{{x}^{2}}+bx-16\] has integral roots is
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

Answer 1

Hint: We know that the roots of the quadratic equation \[a{{x}^{2}}+bx+c=0\]are given by \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] . Also we know that these roots are integer roots when \[b\] is an even integer and its discriminant \[D={{b}^{2}}-4ac\] is a perfect square of an even number.

Complete step-by-step solution:
Now we have to find the number of the integral values of b, such that the equation \[{{x}^{2}}+bx-16\] has integral roots.
As we know that, the quadratic equation \[a{{x}^{2}}+bx+c=0\] has integral roots if \[b\] is an even integer and its discriminant \[D={{b}^{2}}-4ac\] is a perfect square of an even number.
Let us consider, \[{{x}^{2}}+bx-16\] which has its third term equal to \[16\] having factors \[1,2,4,8,16\]
Hence, the possible values of \[b\] are as follows:
\[\begin{align}
  & \Rightarrow b=1-16=-15\text{ and 1}\times \left( -16 \right)=-16 \\
 & \Rightarrow b=16-1=15\text{ and 16}\times \left( -1 \right)=-16 \\
 & \Rightarrow b=2-8=-6\text{ and 2}\times \left( -8 \right)=-16 \\
 & \Rightarrow b=8-2=6\text{ and 8}\times \left( -2 \right)=-16 \\
 & \Rightarrow b=4-4=0\text{ and 4}\times \left( -4 \right)=-16 \\
\end{align}\]
But as stated above \[b\] must be an even integer and the discriminant \[D={{b}^{2}}-4ac\] is a perfect square of an even number.
\[\begin{align}
  & \Rightarrow D={{b}^{2}}-4ac \\
 & \Rightarrow D={{b}^{2}}-4\times 1\times \left( -16 \right) \\
 & \Rightarrow D={{b}^{2}}+64 \\
\end{align}\]
So that \[-6,6\] and \[0\]are only the even integer values of \[b\] for which the discriminant of a given quadratic equation has a perfect square of an even number.
Hence, the number of integral values of \[b\] is 3.
Thus option (2) is the correct option.

Note: In this question students have to note that the value of \[b\] depends on the factors of the third term i.e. 16. Students have to note that in quadratic equations if the third term is negative and second term is positive then the second term can be expressed in the form of subtraction of factors of the third term such that the multiplication of the two factors must be equal to the third term.