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If one root of the equation ax2+bx+c = 0 is three times the other, then b2 : ac =
- a)16 : 1
- b)3 : 16
- c)3 : 1
- d)16 : 3
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If one root of the equationax2+bx+c = 0 is three times the other, then...
Let one root be α then other root will be 3α.
∴ Sum of the roots
∴ Sum of the roots
And Product of the roots 
Equating the coefficients of α2 and subtracing e.q. (ii) from e.q. (i), we get
Most Upvoted Answer
If one root of the equationax2+bx+c = 0 is three times the other, then...
Understanding the Problem
Given the quadratic equation \( ax^2 + bx + c = 0 \), we know that one root is three times the other. Let's denote the roots as \( r \) and \( 3r \).
Using Vieta's Formulas
According to Vieta's formulas:
- The sum of the roots \( r + 3r = 4r \) is equal to \( -\frac{b}{a} \).
- The product of the roots \( r \cdot 3r = 3r^2 \) is equal to \( \frac{c}{a} \).
From these, we can derive two equations:
1. \( 4r = -\frac{b}{a} \) (1)
2. \( 3r^2 = \frac{c}{a} \) (2)
Manipulating the Equations
From equation (1), we can express \( r \) in terms of \( b \) and \( a \):
- \( r = -\frac{b}{4a} \)
Now, substituting this value of \( r \) into equation (2):
- \( 3\left(-\frac{b}{4a}\right)^2 = \frac{c}{a} \)
This simplifies to:
- \( 3 \cdot \frac{b^2}{16a^2} = \frac{c}{a} \)
Multiplying both sides by \( 16a^2 \):
- \( 3b^2 = 16ac \)
Finding the Ratio
Thus, rearranging gives us:
- \( \frac{b^2}{ac} = \frac{16}{3} \)
This leads us to the final ratio:
Conclusion
- Therefore, the ratio \( b^2 : ac \) is \( 16 : 3 \).
Hence, the correct answer is option **D**: \( 16 : 3 \).
Given the quadratic equation \( ax^2 + bx + c = 0 \), we know that one root is three times the other. Let's denote the roots as \( r \) and \( 3r \).
Using Vieta's Formulas
According to Vieta's formulas:
- The sum of the roots \( r + 3r = 4r \) is equal to \( -\frac{b}{a} \).
- The product of the roots \( r \cdot 3r = 3r^2 \) is equal to \( \frac{c}{a} \).
From these, we can derive two equations:
1. \( 4r = -\frac{b}{a} \) (1)
2. \( 3r^2 = \frac{c}{a} \) (2)
Manipulating the Equations
From equation (1), we can express \( r \) in terms of \( b \) and \( a \):
- \( r = -\frac{b}{4a} \)
Now, substituting this value of \( r \) into equation (2):
- \( 3\left(-\frac{b}{4a}\right)^2 = \frac{c}{a} \)
This simplifies to:
- \( 3 \cdot \frac{b^2}{16a^2} = \frac{c}{a} \)
Multiplying both sides by \( 16a^2 \):
- \( 3b^2 = 16ac \)
Finding the Ratio
Thus, rearranging gives us:
- \( \frac{b^2}{ac} = \frac{16}{3} \)
This leads us to the final ratio:
Conclusion
- Therefore, the ratio \( b^2 : ac \) is \( 16 : 3 \).
Hence, the correct answer is option **D**: \( 16 : 3 \).
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If one root of the equationax2+bx+c = 0 is three times the other, thenb2: ac =a)16 : 1b)3 : 16c)3 : 1d)16 : 3Correct answer is option 'D'. 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 one root of the equationax2+bx+c = 0 is three times the other, thenb2: ac =a)16 : 1b)3 : 16c)3 : 1d)16 : 3Correct answer is option 'D'. 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 one root of the equationax2+bx+c = 0 is three times the other, thenb2: ac =a)16 : 1b)3 : 16c)3 : 1d)16 : 3Correct answer is option 'D'. Can you explain this answer?.
If one root of the equationax2+bx+c = 0 is three times the other, thenb2: ac =a)16 : 1b)3 : 16c)3 : 1d)16 : 3Correct answer is option 'D'. 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 one root of the equationax2+bx+c = 0 is three times the other, thenb2: ac =a)16 : 1b)3 : 16c)3 : 1d)16 : 3Correct answer is option 'D'. 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 one root of the equationax2+bx+c = 0 is three times the other, thenb2: ac =a)16 : 1b)3 : 16c)3 : 1d)16 : 3Correct answer is option 'D'. Can you explain this answer?.
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