Location of Roots - Practice Questions & MCQ

Let f(x) = ax² + bx + c , where a,b,c are real numbers and ‘a’ is non-zero number. Let ? and ? be the roots of the equation, and let k be a real number. Then:

1. If both roots of f(x) are less than k then

i) D ≥ 0 (as the real roots may be distinct or equal)

ii) af(k) > 0 (In both cases af(k) is positive, as in the second case if a < 0 then f(k) < 0, so multiplying two -ve values will give us a positive value)

iii)$\mathrm{k}>\frac{-\mathrm{b}}{2 \mathrm{a}}$ since, $\frac{-\mathrm{b}}{2 \mathrm{a}}$ will lie between ? and ?, and ?, ? are less than k so $\frac{-b}{2 a}$ will be less than k.

2. If both roots of f(x) are greater than k

i) D ≥ 0 (as the real roots may be distinct or equal)

ii) af(k) > 0 (In both cases af(k) is positive, as in the second case if a < 0 then f(k) < 0, so multiplying two -ve values will give us a positive value)

iii)$\mathrm{k}<\frac{-\mathrm{b}}{2 \mathrm{a}}$ since $\frac{-\mathrm{b}}{2 \mathrm{a}}$ will lie between ? and $\underline{\underline{\underline{?}}}$, and $\underline{\underline{\underline{?}}}$ ? are greater than k so $\frac{-b}{2 a}$ will be greater than k .

Condition for number k

Let f(x) = ax² + bx + c where a, b, c are real numbers and ‘a’ is a non-zero number. Let ? and ? be the real roots of the function. And let k be any real number. Then:

1. If k lies between the root ? and ?

af(k) < 0.

(As if a < 0 then f(k) > 0. So multiplying one -ve and one +ve value will give us a negative value)

Condition for numbers k₁ and k₂

Let f(x) = ax² + bx + c, where a,b,c are real numbers and ‘a’ isa  non-zero number. Let ? and ? be the real roots of the function. And let $k_1, k_2$  be any two real numbers. Then:

2.        If exactly one root of f(x) lies in between the number  $k_1, k_2$

$f\left(k_1\right) f\left(k_2\right)<0$  as for one value of k, we will have +ve value of f(x) and for other value of k, we will have a -ve value of f(x) (here  ? < ?)

Condition on number $k_1, k_2$

Let f(x) = ax² + bx + c, where a, b, c are real numbers and ‘a’ is non-zero number. Let ? and ? be the roots of the function. And let $k_1, k_2$  be any two real numbers. Then

  1.  If both roots lie between  $k_1, k_2$

i) D ≥ 0 (as the real roots may be distinct or equal)

ii) $\begin{aligned} & \operatorname{af}\left(\mathrm{k}_1\right)>0 \text { and } \mathrm{af}\left(\mathrm{k}_2\right)>0 \\ & \mathrm{k}_1<\frac{-\mathrm{b}}{2 \mathrm{a}}<\mathrm{k}_2, \text { where } \alpha \leq \beta \text { and } \mathrm{k}_1<\mathrm{k}_2\end{aligned}$

2.  If  $k_1, k_2$  lies  between the roots

$\mathrm{af}\left(\mathrm{k}_1\right)<0$ and $\mathrm{af}\left(\mathrm{k}_2\right)<0$