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If the quadratic equation (a2 - b2)x2 + (b2 - c2)x + (c2 - a2) = 0 has equal roots, then which of the following is true?
- a)b2 + c2 = a2
- b)b2 + c2 = 2a2
- c)b2 - c2 = 2a2
- d)a2 = b2 + 2c2
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If the quadratic equation (a2 - b2)x2 + (b2 - c2)x + (c2 - a2) = 0 has...
To find the conditions for the given quadratic equation to have equal roots, we can use the discriminant. The discriminant is the expression within the square root in the quadratic formula, given by b^2 - 4ac. If the discriminant is equal to zero, then the quadratic equation has equal roots.
Let's analyze the given quadratic equation: (a^2 - b^2)x^2 + (b^2 - c^2)x + (c^2 - a^2) = 0
The coefficients of the quadratic equation are:
a = (a^2 - b^2)
b = (b^2 - c^2)
c = (c^2 - a^2)
Using the formula for the discriminant, we have:
Discriminant = b^2 - 4ac
= (b^2 - c^2)^2 - 4(a^2 - b^2)(c^2 - a^2)
= (b^4 - 2b^2c^2 + c^4) - 4(a^2 - b^2)(c^2 - a^2)
= b^4 - 2b^2c^2 + c^4 - 4(a^2c^2 - a^2b^2 - c^2a^2 + b^2c^2)
= b^4 - 2b^2c^2 + c^4 - 4a^2c^2 + 4a^2b^2 + 4c^2a^2 - 4b^2c^2
= b^4 + c^4 - 2b^2c^2 - 4a^2c^2 + 4a^2b^2 + 4c^2a^2 - 4b^2c^2
Simplifying further, we get:
Discriminant = b^4 + c^4 - 6b^2c^2 + 4a^2b^2 + 4a^2c^2
For the quadratic equation to have equal roots, the discriminant should be equal to zero. Therefore, we have the equation:
b^4 + c^4 - 6b^2c^2 + 4a^2b^2 + 4a^2c^2 = 0
Now, let's examine the given options:
A) b^2 - c^2 = 2a^2
B) b^2 - c^2 = 2a^2
C) b^2 - c^2 = 2a^2
D) a^2 = b^2 - 2c^2
Comparing the options with the derived equation, we can see that option B matches exactly. Therefore, the correct answer is option B: b^2 - c^2 = 2a^2.
It is important to note that this solution can be verified by substituting the given values of a, b, and c into the equation and checking if it satisfies the condition for equal roots.
Let's analyze the given quadratic equation: (a^2 - b^2)x^2 + (b^2 - c^2)x + (c^2 - a^2) = 0
The coefficients of the quadratic equation are:
a = (a^2 - b^2)
b = (b^2 - c^2)
c = (c^2 - a^2)
Using the formula for the discriminant, we have:
Discriminant = b^2 - 4ac
= (b^2 - c^2)^2 - 4(a^2 - b^2)(c^2 - a^2)
= (b^4 - 2b^2c^2 + c^4) - 4(a^2 - b^2)(c^2 - a^2)
= b^4 - 2b^2c^2 + c^4 - 4(a^2c^2 - a^2b^2 - c^2a^2 + b^2c^2)
= b^4 - 2b^2c^2 + c^4 - 4a^2c^2 + 4a^2b^2 + 4c^2a^2 - 4b^2c^2
= b^4 + c^4 - 2b^2c^2 - 4a^2c^2 + 4a^2b^2 + 4c^2a^2 - 4b^2c^2
Simplifying further, we get:
Discriminant = b^4 + c^4 - 6b^2c^2 + 4a^2b^2 + 4a^2c^2
For the quadratic equation to have equal roots, the discriminant should be equal to zero. Therefore, we have the equation:
b^4 + c^4 - 6b^2c^2 + 4a^2b^2 + 4a^2c^2 = 0
Now, let's examine the given options:
A) b^2 - c^2 = 2a^2
B) b^2 - c^2 = 2a^2
C) b^2 - c^2 = 2a^2
D) a^2 = b^2 - 2c^2
Comparing the options with the derived equation, we can see that option B matches exactly. Therefore, the correct answer is option B: b^2 - c^2 = 2a^2.
It is important to note that this solution can be verified by substituting the given values of a, b, and c into the equation and checking if it satisfies the condition for equal roots.
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If the quadratic equation (a2 - b2)x2 + (b2 - c2)x + (c2 - a2) = 0 has...
Observe the coefficient of each term and guess one root.
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If the quadratic equation (a2 - b2)x2 + (b2 - c2)x + (c2 - a2) = 0 hasequal roots, then which of the following is true?a)b2 +c2 = a2b)b2 + c2 = 2a2c)b2 - c2 = 2a2d)a2 = b2+ 2c2Correct answer is option 'B'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If the quadratic equation (a2 - b2)x2 + (b2 - c2)x + (c2 - a2) = 0 hasequal roots, then which of the following is true?a)b2 +c2 = a2b)b2 + c2 = 2a2c)b2 - c2 = 2a2d)a2 = b2+ 2c2Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the quadratic equation (a2 - b2)x2 + (b2 - c2)x + (c2 - a2) = 0 hasequal roots, then which of the following is true?a)b2 +c2 = a2b)b2 + c2 = 2a2c)b2 - c2 = 2a2d)a2 = b2+ 2c2Correct answer is option 'B'. Can you explain this answer?.
If the quadratic equation (a2 - b2)x2 + (b2 - c2)x + (c2 - a2) = 0 hasequal roots, then which of the following is true?a)b2 +c2 = a2b)b2 + c2 = 2a2c)b2 - c2 = 2a2d)a2 = b2+ 2c2Correct answer is option 'B'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If the quadratic equation (a2 - b2)x2 + (b2 - c2)x + (c2 - a2) = 0 hasequal roots, then which of the following is true?a)b2 +c2 = a2b)b2 + c2 = 2a2c)b2 - c2 = 2a2d)a2 = b2+ 2c2Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the quadratic equation (a2 - b2)x2 + (b2 - c2)x + (c2 - a2) = 0 hasequal roots, then which of the following is true?a)b2 +c2 = a2b)b2 + c2 = 2a2c)b2 - c2 = 2a2d)a2 = b2+ 2c2Correct answer is option 'B'. Can you explain this answer?.
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