Name: Introduction to Algebra
Uploaded: 2019-07-19
Duration: 11 min 18 s
Description: In this video, we have discussed basic concepts of Algebra. Having a clear idea of the concept of this chapter will helps you to solve problems which are related to this topic.

Question

You are given that the equation

x^{3} + a x^{2} + b x + c = 0

has three real roots and that these roots are consecutive terms in an arithmetic progression. Show that

2 a^{3} + 27 c = 9 a b

.

Accepted Answer

Given cubic equation $x^{3} + a x^{2} + b x + c = 0$ has three roots in arithmetic progression.

Let these roots are $(n - m), n$ and $(n + m)$ .

We know for cubic equation $p x^{3} + q x^{2} + r x + s = 0$ has three roots $x_{1}, x_{2}, x_{3}$ , then there is three relations between roots,

$x_{1} + x_{2} + x_{3} = - \frac{q}{p} x_{1} x_{2} + x_{2} x_{3} + x_{3} x_{1} = \frac{r}{p} x_{1} x_{2} x_{3} = - \frac{s}{p}$

Now for cubic equation $x^{3} + a x^{2} + b x + c = 0$ ,

$n - m + n + n + m = - a \Rightarrow 3 n = - a \Rightarrow n = - \frac{a}{3} . . . . . . . (i) e q .$

$(n - m) n + n (n + m) + (n + m) (n - m) = b \Rightarrow 3 n^{2} - m^{2} = b \Rightarrow m^{2} = 3 n^{2} - b . . . . . . . (ii) e q .$

$(n - m) n (n + m) = - c \Rightarrow n^{3} - n m^{2} = - c \Rightarrow n^{3} - n (3 n^{2} - b) = - c \Rightarrow - 2 n^{3} + n b = - c \Rightarrow - 2 {(\frac{- a}{3})}^{3} + (\frac{- a}{3}) b = - c \Rightarrow \frac{2 a^{3}}{27} - \frac{a b}{3} = - c \Rightarrow 2 a^{3} - 9 a b = - 27 c \Rightarrow 2 a^{3} + 27 c = 9 a b$

Hence proved.

You are given that the equation $x^{3} + a x^{2} + b x + c = 0$ has three real roots and that these roots are consecutive terms in an arithmetic progression. Show that $2 a^{3} + 27 c = 9 a b$ .

Important Questions on Algebra

Learn and Prepare for any exam you want !

You are given that the equation x3+ax2+bx+c=0 has three real roots and that these roots are consecutive terms in an arithmetic progression. Show that 2a3+27c=9ab.

Important Questions on Algebra

Learn and Prepare for any exam you want !

You are given that the equation $x^{3} + a x^{2} + b x + c = 0$ has three real roots and that these roots are consecutive terms in an arithmetic progression. Show that $2 a^{3} + 27 c = 9 a b$ .