If the roots of x² + px + 12 = 0 are in the ratio 1:3 and both are negative, what is p?
Answer and explanation
Correct answer: 8
Let the two roots in the ratio 1:3 be k and 3k. Their product is 3k². Since the constant term of x² + px + 12 = 0 is 12, Vieta’s relation gives 3k² = 12, so k² = 4 and k = ±2. Both roots are specified to be negative, so k = −2. The roots are therefore −2 and −6, whose sum is −8. For a monic quadratic x² + px + 12 = 0, the sum of the roots is −p. Thus −p = −8, giving p = 8. Option A is correct. Choosing positive roots would violate the stated condition and lead to the wrong sign.
Frequently asked questions
What is the correct answer to this question?
8
Why is this the correct answer?
Let the two roots in the ratio 1:3 be k and 3k. Their product is 3k². Since the constant term of x² + px + 12 = 0 is 12, Vieta’s relation gives 3k² = 12, so k² = 4 and k = ±2. Both roots are specified to be negative, so k = −2. The roots are therefore −2 and −6, whose sum is −8. For a monic quadratic x² + px + 12 = 0, the sum of the roots is −p. Thus −p = −8, giving p = 8. Option A is correct. Choosing positive roots would violate the stated condition and lead to the wrong sign.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Roots of a Quadratic Equation.
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