For complex roots of a quadratic equation \(ax^2 + bx + c = 0\), the discriminant \(b^2 - 4ac\) must be negative. In your case, \(a = 3\), \(b = -12\), and \(c = k\). So, the condition for complex roots is \((-12)^2 - 4(3)(k) < 0\). Simplify this inequality to find the range of \(k\).
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Answer:
For the quadratic equation 3x² - 12x + k = 0 to have complex roots, the discriminant (b² - 4ac) must be negative, where a = 3, b = -12, and c = k.
So, the discriminant D = (-12)² - 4(3)(k) = 144 - 12k.
For complex roots, D < 0. Therefore:
144 - 12k < 0
Now, solve for k:
12k > 144
k > 144 / 12
k > 12
So, the range of k for the roots of 3x² - 12x + k = 0 to be complex is k > 12.
Step-by-step explanation:
Complex Roots(√) Range :-
For complex roots of a quadratic equation \(ax^2 + bx + c = 0\), the discriminant \(b^2 - 4ac\) must be negative. In your case, \(a = 3\), \(b = -12\), and \(c = k\). So, the condition for complex roots is \((-12)^2 - 4(3)(k) < 0\). Simplify this inequality to find the range of \(k\).