Question

4fx²-(m-3)x+m≥0 (m∈R) Find 'm' if (i) Both roots are +ve. (ii) Both roots are -ve. (iii) Roots are of opposite sign. (iv) Atleast one root is +ve. (v) Atleast one root is -ve.

• A. (i) Both roots are +ve: $m \in [11+4\sqrt{7}, \infty)$
• B. (ii) Both roots are -ve: $m \in (0, 11-4\sqrt{7}]$
• C. (iii) Roots are of opposite sign: $m \in (-\infty, 0)$
• D. (iv) Atleast one root is +ve: $m \in (-\infty, 0) \cup [11+4\sqrt{7}, \infty)$
• E. (v) Atleast one root is -ve: $m \in (-\infty, 11-4\sqrt{7}]$

Answer 1

(i) Both roots are +ve: $m \in [11+4\sqrt{7}, \infty)$ (ii) Both roots are -ve: $m \in (0, 11-4\sqrt{7}]$ (iii) Roots are of opposite sign: $m \in (-\infty, 0)$ (iv) Atleast one root is +ve: $m \in (-\infty, 0) \cup [11+4\sqrt{7}, \infty)$ (v) Atleast one root is -ve: $m \in (-\infty, 11-4\sqrt{7}]$

Answer 2

The conditions for the nature of roots of a quadratic equation $ax^2+bx+c=0$ are determined by its discriminant ($\Delta = b^2-4ac$), sum of roots ($S = -b/a$), and product of roots ($P = c/a$). For real roots, $\Delta \ge 0$.

• Both roots positive: $\Delta \ge 0, S > 0, P > 0$.
• Both roots negative: $\Delta \ge 0, S < 0, P > 0$.
• Roots of opposite sign: $P < 0$ (implies $\Delta > 0$).
• At least one root positive: (roots of opposite sign) OR (both roots positive).
• At least one root negative: (roots of opposite sign) OR (both roots negative or zero, and at least one strictly negative). Applying these conditions to $4x^2 - (m-3)x + m = 0$ with $a=4, b=-(m-3), c=m$, we find $\Delta = m^2-22m+9$, $S=(m-3)/4$, $P=m/4$. Solving the inequalities for $m$ yields the respective intervals.