If one root of the equation ax^2 + bx + c = 0 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $n\alpha$.According to the relationship between roots and coefficients

Vedclass · Accepted Answer

$n{b^2} = ac{(n + 1)^2}$

Vedclass · Answer

(B) Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $n\alpha$.According to the relationship between roots and coefficients:Sum of roots: $\alpha + n\alpha = -\frac{b}{a} \implies \alpha(n + 1) = -\frac{b}{a} \implies \alpha = -\frac{b}{a(n + 1)}$ ... $(i)$Product of roots: $\alpha \cdot n\alpha = \frac{c}{a} \implies n\alpha^2 = \frac{c}{a} \implies \alpha^2 = \frac{c}{na}$ ... (ii)Substituting the value of $\alpha$ from equation $(i)$ into equation (ii):$\left(-\frac{b}{a(n + 1)}\right)^2 = \frac{c}{na}$$\frac{b^2}{a^2(n + 1)^2} = \frac{c}{na}$Multiplying both sides by $a^2(n + 1)^2$:$b^2 = \frac{c \cdot a^2(n + 1)^2}{na}$$b^2 = \frac{ac(n + 1)^2}{n}$$n{b^2} = ac{(n + 1)^2}$.

If one root of the equation $ax^2 + bx + c = 0$ is $n$ times the other root,then:

Similar Questions

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}+12x+36=0$
$II.$ $y^{2}+15y+56=0$

The equation formed by decreasing each root of $ax^2 + bx + c = 0$ by $1$ is $2x^2 + 8x + 2 = 0$. Then:

If $\alpha \ne \beta$ but $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$,then the equation whose roots are $\alpha/\beta$ and $\beta/\alpha$ is

The real number $k$ for which the equation $2x^2 + 3x + k = 0$ has two distinct real roots in $[0, 1]$:

If $3p^2 = 5p + 2$ and $3q^2 = 5q + 2$ where $p \ne q$,then the equation whose roots are $3p - 2q$ and $3q - 2p$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module