For what values o f c in the equation 2x2 - (c3 + 8c - l)x + c2 - 4c = 0 the roots of the equation would be opposite in signs?a)c € (0 ,4 )b)c € ( - 4 , 0 )c)c € (0 ,3 )d)c € ( - 4 , 4 )Correct answer is option 'A'. Can you explain this answer?

Question

For what values o f c in the equation 2x2 - (c3 + 8c - l)x + c2 - 4c = 0 the roots of the equation would be opposite in signs?a)c &#8364; (0 ,4 )b)c &#8364; ( - 4 , 0 )c)c &#8364; (0 ,3 )d)c &#8364; ( - 4 , 4 )Correct answer is option 'A'. Can you explain this answer?

Answer

Solution:

The given quadratic equation is 2x² - (c³ - 8c - 1)x - c² + 4c = 0.

Let α and β be the roots of the given quadratic equation.

We know that the sum and product of the roots of a quadratic equation ax² + bx + c = 0 are given by -b/a and c/a respectively.

∴ α + β = (c³ - 8c - 1)/2 and αβ = (c² - 4c)/2

Given that the roots of the equation are opposite in signs.

∴ α and β have opposite signs, i.e., one is positive and the other is negative.

Case 1: α > 0 and β < />

We have αβ < />

i.e., (c² - 4c)/2 < />

⇒ c(c - 4) < />

⇒ c ∈ (0, 4)

Case 2: α < 0='' and='' β='' /> 0

We have αβ < />

i.e., (c² - 4c)/2 < />

⇒ c(c - 4) > 0

⇒ c ∈ (-∞, 0) ∪ (4, ∞)

Therefore, the value of c for which the roots of the equation would be opposite in signs is (0, 4).

Hence, option A is the correct answer.

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