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The equation x3 + (2r + 1)x2 + (4r - 1)x + 2 = 0 has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r is
Correct answer is '2'. Can you explain this answer?
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The equation x3 + (2r + 1)x2 + (4r - 1)x + 2 = 0 has -2 as one of the ...
Given Information:
- The equation x3 + (2r + 1)x2 + (4r - 1)x + 2 = 0 has -2 as one of the roots.
- The other two roots are real.
Explanation:
Roots of a Cubic Equation:
- By Vieta's formulas, the sum of the roots of a cubic equation ax3 + bx2 + cx + d = 0 is equal to -b/a.
- Since one root is given as -2 and the other two roots are real, the sum of all roots must be real.
Sum of Roots:
- The sum of roots of the given equation is (-2) + α + β = -(2r + 1).
Product of Roots:
- The product of roots of the given equation is -2αβ = 2.
Minimum Value of r:
- From the sum of roots, we get: α + β = -(2r + 1) + 2.
- From the product of roots, we get: αβ = -1.
- By solving these two equations, we get the values of α and β.
- Substitute these values in the sum of roots equation to find the minimum value of r.
Minimum Value of r:
- By solving, we get r = 2.
Therefore, the minimum possible non-negative integer value of r is 2.
- The equation x3 + (2r + 1)x2 + (4r - 1)x + 2 = 0 has -2 as one of the roots.
- The other two roots are real.
Explanation:
Roots of a Cubic Equation:
- By Vieta's formulas, the sum of the roots of a cubic equation ax3 + bx2 + cx + d = 0 is equal to -b/a.
- Since one root is given as -2 and the other two roots are real, the sum of all roots must be real.
Sum of Roots:
- The sum of roots of the given equation is (-2) + α + β = -(2r + 1).
Product of Roots:
- The product of roots of the given equation is -2αβ = 2.
Minimum Value of r:
- From the sum of roots, we get: α + β = -(2r + 1) + 2.
- From the product of roots, we get: αβ = -1.
- By solving these two equations, we get the values of α and β.
- Substitute these values in the sum of roots equation to find the minimum value of r.
Minimum Value of r:
- By solving, we get r = 2.
Therefore, the minimum possible non-negative integer value of r is 2.
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The equation x3 + (2r + 1)x2 + (4r - 1)x + 2 = 0 has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r isCorrect answer is '2'. Can you explain this answer?
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