Topics

General Knowledge
General Science
General English
Aptitude
General Computer Science
General Intellingence and Reasoning
Current Affairs
Exams
Elementary Mathematics
English Literature

Both the roots of the given equation. \( \Large \left(x-a\right) \left(x-b\right)+ \left(x-b\right) \left(x-c\right)+ \left(x-c\right) \left(x-a\right)=0 \) are always

A) positive
B) negative
C) real
D) imaginary

Correct Answer:



C) real

Description for Correct answer:
Given equation:

\( \Large \left(x-a\right) \left(x-b\right)+ \left(x-b\right) \left(x-c\right)+ \left(x-c\right) \left(x-a\right)=0 \)

\( \Large 3x^{2}-2 \left(a+b+c\right)x+ \left(ab+bc+ca\right)=0 \)

Now, \( \Large BZ-4AC=4\left[ \left(a+b+c+\right)^{2}-3 \left(ab+bc+ca\right) \right] \)

= \( \Large 4 \left(a^{2}+b^{2}+c^{2}-ab-bc-ac\right) \)

= \( \Large 2\{ \left(a-b\right)^{2}+ \left(b-c\right)^{2}+ \left(c-a\right)^{2} \}\ge 0 \)

Hence, both roots are always real.

Part of solved Quadratic Equations questions and answers : >> Elementary Mathematics >> Quadratic Equations

Comments

Similar Questions

1). If the difference of roots of the equation \( \Large x^{2}-bx+c=0 \) be 1, then;

A). \( \Large b^{2}-4c-1=0 \)

B). \( \Large b^{2}-4c=0 \)

C). \( \Large b^{2}-4c+1=0 \)

D). \( \Large b^{2}+4c-1=0 \)

2). If \( \Large 2+i\sqrt{3} \) is a root of the equation \( \Large x^{2}+px+q=0 \) where p and q are real, then \( \Large p, q \) is equal to:

A). (-4, 7)

B). (4,-7)

C). (4, 7)

D). (-4, -7)

3). If \( \Large x=\sqrt{1+\sqrt{1+\sqrt{1}+.....}} \), the x is equal to

A). \( \Large \frac{1+\sqrt{5}}{2} \)

B). \( \Large \frac{1-\sqrt{5}}{2} \)

C). \( \Large \frac{1\pm \sqrt{5}}{2} \)

D). none of these .

4). If \( \Large \alpha , \beta \) are the roots of equation \( \Large ax^{2}+bx+c=0 \), then \( \Large \frac{ \alpha }{ \alpha \beta +b}+\frac{ \beta }{ \alpha x+b} \) is equal to

A). 2/a

B). 2/b

C). 2/c

D). -2/a

5). If the roots of the equation \( \Large \frac{ \alpha }{x- \alpha }+\frac{ \beta }{x- \beta }=1 \) be equal in magnitude but opposite in sign, then \( \Large \alpha + \beta \) is equal to:

A). 0

B). 1

C). 2

D). none of these

6). If the roots of the equation \( \Large px^{2}+2qx+r=0 \) and \( \Large qx^{2}-2\sqrt{pr}x+q=0 \) be real, then:

A). \( \Large p=q \)

B). \( \Large q^{2}=pr \)

C). \( \Large p^{2}=qr \)

D). \( \Large r^{2}=pq \)

7). If one root of the quadratic equation \( \Large ax^{2}+bx+c=0 \) is equal to nth power of the other root, then the value of \( \Large \left(ac^{n}\right)^{\frac{1}{n+1}} + \left(a^{n}c\right)^{\frac{1}{n+1}} \) is equal to

A). b

B). -b

C). \( \Large b^{1/n+1} \)

D). \( \Large -b^{1/n+1} \)

8). The quadratic in t, such that AM of its roots is A and GM is G is:

A). \( \Large t^{2}-2At+G^{2}=0 \)

B). \( \Large t^{2}-2At-G^{2}=0 \)

C). \( \Large t^{2}+2At+G^{2}=0 \)

D). none of these.

9). If \( \Large n^{2}px+1 \) is a factor of expression \( \Large ax^{3}+bx+c \) then:

A). \( \Large a^{2}+c^{2}=-ab \)

B). \( \Large a^{2}-c^{2}=-ab \)

C). \( \Large a^{2}-c^{2}=ab \)

D). none of these

10). If \( \Large \sqrt{3x^{2}-7x-30} + \sqrt{2x^{2}-7x-5} = x+5 \), then x is equal to:

A). 2

B). 3

C). 6

D). 5