Banking Exams Exam > Banking Exams Questions > If the roots of the equation(a2+ b2)x2−...
Start Learning for Free
If the roots of the equation (a2 + b2)x2 − 2b(a + c)x + (b2+c2) = 0 are equal then
- a)2b = ac
- b)b2 = ac
- c)b = 2ac/(a + c)
- d)b = ac
- e)b = 2ac
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0...
(a2 + b2)x2 − 2b(a + c)x + (b2+c2) = 0
Roots are real and equal ∴ D = 0
D = b2 − 4ac = 0
⇒ [−2b(a+c)]2 − 4(a2 + b2)(b2 + c2) = 0
⇒ b2(a2 + c2 + 2ac) −(a2b2 + a2c2 + b4 + c2c2) = 0
⇒ b2a2 + b2c2 + 2acb2 − a2b2 − a2c2 − b4 − b2c2 = 0
⇒ 2acb2 − a2c2 − 2acb2 = 0
⇒ (b2 − ac)2 = 0
⇒ b2 = ac
Roots are real and equal ∴ D = 0
D = b2 − 4ac = 0
⇒ [−2b(a+c)]2 − 4(a2 + b2)(b2 + c2) = 0
⇒ b2(a2 + c2 + 2ac) −(a2b2 + a2c2 + b4 + c2c2) = 0
⇒ b2a2 + b2c2 + 2acb2 − a2b2 − a2c2 − b4 − b2c2 = 0
⇒ 2acb2 − a2c2 − 2acb2 = 0
⇒ (b2 − ac)2 = 0
⇒ b2 = ac
Free Test
FREE
| Start Free Test |
Community Answer
If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0...
We can rearrange the given equation to get:
(a^2 - b^2)x^2 + 2abx - 3ab = 0
The discriminant of this quadratic equation is:
D = (2ab)^2 - 4(a^2 - b^2)(-3ab)
= 4a^2b^2 + 12a^3b - 12ab^3
We want to find the conditions for which D is negative, which will imply that the quadratic equation has no real roots.
4a^2b^2 + 12a^3b - 12ab^3 < />
Dividing both sides by 4ab, we get:
ab + 3a^2 - 3b^2 < />
We can rewrite this inequality as:
ab < 3b^2="" -="" />
Now, we need to consider two cases:
1. b > 0: In this case, we can divide both sides by b^2 and get:
a/b < 3="" -="" />
Letting x = a/b, we can rewrite this as:
x^2 + 3x - 3 < />
This is a quadratic inequality that has solutions for:
-3 - sqrt(21) < x="" />< -3="" +="" />
Since x = a/b, this implies that:
a/b < -3="" -="" sqrt(21)="" or="" a/b="" /> -3 + sqrt(21)
2. b < 0:="" in="" this="" case,="" we="" can="" divide="" both="" sides="" by="" -b^2="" and="" />
-a/b < 3="" -="" />
Letting x = a/b, we can rewrite this as:
x^2 - 3x - 3 < />
This is a quadratic inequality that has solutions for:
(3 - sqrt(21)) < x="" />< (3="" +="" />
Since x = a/b, this implies that:
a/b < 3="" -="" sqrt(21)="" or="" a/b="" /> 3 + sqrt(21)
Therefore, the roots of the given quadratic equation are real if and only if:
a/b < -3="" -="" sqrt(21)="" or="" a/b="" /> -3 + sqrt(21) or a/b < 3="" -="" sqrt(21)="" or="" a/b="" /> 3 + sqrt(21)
(a^2 - b^2)x^2 + 2abx - 3ab = 0
The discriminant of this quadratic equation is:
D = (2ab)^2 - 4(a^2 - b^2)(-3ab)
= 4a^2b^2 + 12a^3b - 12ab^3
We want to find the conditions for which D is negative, which will imply that the quadratic equation has no real roots.
4a^2b^2 + 12a^3b - 12ab^3 < />
Dividing both sides by 4ab, we get:
ab + 3a^2 - 3b^2 < />
We can rewrite this inequality as:
ab < 3b^2="" -="" />
Now, we need to consider two cases:
1. b > 0: In this case, we can divide both sides by b^2 and get:
a/b < 3="" -="" />
Letting x = a/b, we can rewrite this as:
x^2 + 3x - 3 < />
This is a quadratic inequality that has solutions for:
-3 - sqrt(21) < x="" />< -3="" +="" />
Since x = a/b, this implies that:
a/b < -3="" -="" sqrt(21)="" or="" a/b="" /> -3 + sqrt(21)
2. b < 0:="" in="" this="" case,="" we="" can="" divide="" both="" sides="" by="" -b^2="" and="" />
-a/b < 3="" -="" />
Letting x = a/b, we can rewrite this as:
x^2 - 3x - 3 < />
This is a quadratic inequality that has solutions for:
(3 - sqrt(21)) < x="" />< (3="" +="" />
Since x = a/b, this implies that:
a/b < 3="" -="" sqrt(21)="" or="" a/b="" /> 3 + sqrt(21)
Therefore, the roots of the given quadratic equation are real if and only if:
a/b < -3="" -="" sqrt(21)="" or="" a/b="" /> -3 + sqrt(21) or a/b < 3="" -="" sqrt(21)="" or="" a/b="" /> 3 + sqrt(21)
|
Explore Courses for Banking Exams exam
|
|
Similar Banking Exams Doubts
Top Courses for Banking ExamsView all
If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer?
Question Description
If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer? for Banking Exams 2025 is part of Banking Exams preparation. The Question and answers have been prepared according to the Banking Exams exam syllabus. Information about If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Banking Exams 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer?.
If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer? for Banking Exams 2025 is part of Banking Exams preparation. The Question and answers have been prepared according to the Banking Exams exam syllabus. Information about If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Banking Exams 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer?.
Solutions for If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Banking Exams.
Download more important topics, notes, lectures and mock test series for Banking Exams Exam by signing up for free.
Here you can find the meaning of If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If the roots of the equation(a2+ b2)x2− 2b(a + c)x + (b2+c2) = 0are equal thena)2b = acb)b2= acc)b = 2ac/(a + c)d)b = ace)b = 2acCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice Banking Exams tests.
|
|
Explore Courses for Banking Exams exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up
within 7 days!
within 7 days!
Takes less than 10 seconds to signup