# If $\alpha $ and $\beta $, $\alpha $ and $\gamma $, $\alpha $ and $\delta $ are the roots of the equations $a{x^2} + 2bx + c = 0$, $2b{x^2} + cx + a = 0$ and $c{x^2} + ax + 2b = 0$ respectively, where $a,b,c$ are positive real numbers, then $\alpha + {\alpha ^2} = $

A. $ - 1$

B. $0$

C. $abc$

D. $a + 2b + c$

E. $abc$

Answer

Verified

121.5k+ views

**Hint:**Use the fact that $\alpha $ is a root for all the equations. Substitute $x = \alpha $ in all the equations and all three equations to get a single equation. Write the obtained equation as a product of two factors and use the fact that $a,b,c$ are positive real numbers.

**Formula used:**If $\phi $ is a root of the polynomial $f(x) = a{x^2} + bx + c$, then $f(\phi ) = a{\phi ^2} + b\phi + c = 0$

**Complete step-by-step solution:**

Let $f(x) = a{\alpha ^2} + 2b\alpha + c$, $g(x) = 2b{\alpha ^2} + c\alpha + a$ and $h(x) = c{\alpha ^2} + a\alpha + 2b$

Since $\alpha $ is a root for all the equations, $f(\alpha ) = 0$, $g(\alpha ) = 0$ and $h(\alpha ) = 0$

$a{\alpha ^2} + 2b\alpha + c = 0$

\[2b{\alpha ^2} + c\alpha + a = 0\]

$c{\alpha ^2} + a\alpha + 2b = 0$

Adding the three equations above we get,

$a{\alpha ^2} + 2b\alpha + c + 2b{\alpha ^2} + c\alpha + a + c{\alpha ^2} + a\alpha + 2b = 0$

$\left( {a + 2b + c} \right)\left( {{\alpha ^2} + \alpha + 1} \right) = 0$

$a + 2b + c \ne 0$. Therefore,

\[{\alpha ^2} + \alpha + 1 = 0\]

${\alpha ^2} + \alpha = - 1$

**Therefore, the correct answer is Option A. $ - 1$.**

**Note:**If three numbers are positive numbers then their sum must also be a positive number and therefore, they cannot be 0. That is why $a + 2b + c \ne 0$. Alternatively, we can say the same by writing three inequalities and adding them as follows: $a > 0$, $2b > 0$, $c > 0$. Adding the three inequalities we get $a + 2b + c > 0$.

Last updated date: 30th Sep 2023

•

Total views: 121.5k

•

Views today: 2.21k

Recently Updated Pages

What do you mean by public facilities

Slogan on Noise Pollution

Paragraph on Friendship

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

What is the Full Form of ILO, UNICEF and UNESCO

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

What is the past tense of read class 10 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

10 examples of evaporation in daily life with explanations

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE

What is the past participle of wear Is it worn or class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

What is the difference between anaerobic aerobic respiration class 10 biology CBSE