If ax^2 + bx + c = 0 and bx^2 + cx + a = 0 ha | vedclass

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

Question

(B) Let $\alpha$ be the common root of the equations $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$.Then,$a\alpha^2 + b\alpha + c = 0$ and $b\alpha^2 + c

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(B) Let $\alpha$ be the common root of the equations $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$.Then,$a\alpha^2 + b\alpha + c = 0$ and $b\alpha^2 + c\alpha + a = 0$.Using the method of cross-multiplication for these two equations:$\frac{\alpha^2}{b(a) - c(c)} = \frac{\alpha}{c(b) - a(a)} = \frac{1}{a(c) - b(b)}$$\frac{\alpha^2}{ab - c^2} = \frac{\alpha}{bc - a^2} = \frac{1}{ac - b^2}$From the first and third parts: $\alpha^2 = \frac{ab - c^2}{ac - b^2}$From the second and third parts: $\alpha = \frac{bc - a^2}{ac - b^2}$Since $\alpha^2 = (\alpha)^2$,we have:$\frac{ab - c^2}{ac - b^2} = \left( \frac{bc - a^2}{ac - b^2} ight)^2$$(ab - c^2)(ac - b^2) = (bc - a^2)^2$$a^2bc - ab^3 - ac^3 + b^2c^2 = b^2c^2 - 2a^2bc + a^4$$a^4 + ab^3 + ac^3 = 3a^2bc$Dividing both sides by $abc$ (since $a, b, c 
eq 0$):$\frac{a^3}{bc} + \frac{b^2}{c} + \frac{c^2}{b} = 3$ (This is not the target expression).Actually,dividing $a^4 + ab^3 + ac^3 = 3a^2bc$ by $abc$ gives $\frac{a^3}{bc} + \frac{b^2}{c} + \frac{c^2}{b} = 3$. Wait,let's re-evaluate: $a(a^3 + b^3 + c^3) = 3a^2bc \Rightarrow \frac{a^3 + b^3 + c^3}{abc} = 3$.

If $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root and $a, b, c$ are non-zero real numbers,then $\frac{a^3 + b^3 + c^3}{abc} = $

Similar Questions

If $x-\frac{1}{x}=\sqrt{21},$ then the value of $\left(x^{2}+\frac{1}{x^{2}}\right)\left(x+\frac{1}{x}\right)$ will be

If $x^{4} + \frac{1}{x^{4}} = 119$,then the value of $x^{3} - \frac{1}{x^{3}}$ is

If $a + b + c = 0$,then the roots of the equation $4ax^2 + 3bx + 2c = 0$ are

Solve the given two equations and select the correct option.
$I.$ $x^{2} + 42 = 13x$
$II.$ $y = \sqrt[4]{1296}$

The number of integral values of $m$ for which the equation $(1 + m^2) x^2 - 2(1 + 3m) x + (1 + 8m) = 0$ has no real root is

Vedclass Test Series

Exam Paper Generator

Online Exam Module