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For which one of the following values of k, the equation 2x3 + 3x2 − 12x − k = 0 has three distinct real roots?
- a)16
- b)20
- c)26
- d)31
Correct answer is option 'B'. Can you explain this answer?
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For which one of the following values of k, the equation 2x3 + 3x2 &mi...
Correct answer is option 'A'. Let me know if you still have doubt :)
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For which one of the following values of k, the equation 2x3 + 3x2 &mi...
Identifying the Correct Value of k
To find the value of k for which the given equation has three distinct real roots, we need to use the discriminant of the cubic equation.
Discriminant of a Cubic Equation
The discriminant of a cubic equation ax^3 + bx^2 + cx + d = 0 is given by Δ = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2.
Using the Discriminant to Determine Number of Roots
For a cubic equation to have three distinct real roots, the discriminant must be positive.
Given Equation
2x^3 + 3x^2 - 12x - k = 0
Applying the Values to the Discriminant
For this equation, a = 2, b = 3, c = -12, and d = -k.
Δ = 18*2*(-12)*(-k) - 4*3^3*(-k) + 3^2*(-12)^2 - 4*2*(-12)^3 - 27*2^2(-k)^2
Δ = 432k + 108k + 432 - 4608 + 216k^2
Δ = 324k + 432 + 216k^2 - 4608
Δ = 216k^2 + 324k - 4176
Finding the Values of k
For the equation to have three distinct real roots, the discriminant Δ > 0.
216k^2 + 324k - 4176 > 0
Solving this quadratic inequality, we find that k > 20 or k < />
Therefore, the correct value of k for which the equation has three distinct real roots is k = 20, which is option B.
To find the value of k for which the given equation has three distinct real roots, we need to use the discriminant of the cubic equation.
Discriminant of a Cubic Equation
The discriminant of a cubic equation ax^3 + bx^2 + cx + d = 0 is given by Δ = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2.
Using the Discriminant to Determine Number of Roots
For a cubic equation to have three distinct real roots, the discriminant must be positive.
Given Equation
2x^3 + 3x^2 - 12x - k = 0
Applying the Values to the Discriminant
For this equation, a = 2, b = 3, c = -12, and d = -k.
Δ = 18*2*(-12)*(-k) - 4*3^3*(-k) + 3^2*(-12)^2 - 4*2*(-12)^3 - 27*2^2(-k)^2
Δ = 432k + 108k + 432 - 4608 + 216k^2
Δ = 324k + 432 + 216k^2 - 4608
Δ = 216k^2 + 324k - 4176
Finding the Values of k
For the equation to have three distinct real roots, the discriminant Δ > 0.
216k^2 + 324k - 4176 > 0
Solving this quadratic inequality, we find that k > 20 or k < />
Therefore, the correct value of k for which the equation has three distinct real roots is k = 20, which is option B.
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For which one of the following values of k, the equation 2x3 + 3x2 − 12x − k= 0has three distinct real roots?a)16b)20c)26d)31Correct answer is option 'B'. Can you explain this answer?
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