it is known that b^2 - 4ac is a real root of equation ax^2 + bx + c = 0 (a=/ 0),then ( ) is true.
(A) ab> = 1/8
(B) ab <= 1/8
(C) ab >= 1/4
(D) ab <= 1/4
please show your solution :))
To find the answer to this question, we need to use the given information that b^2 - 4ac is a real root of the equation ax^2 + bx + c = 0.
The discriminant of a quadratic equation is given by b^2 - 4ac. If the discriminant is positive, the quadratic equation will have two real roots. If the discriminant is zero, the quadratic equation will have one real root. If the discriminant is negative, the quadratic equation will have no real roots.
Given that b^2 - 4ac is a real root of the equation, it means that the discriminant (b^2 - 4ac) is zero.
Let's equate the discriminant to zero and solve for it:
b^2 - 4ac = 0
Now, let's rearrange the equation:
b^2 = 4ac
Divide both sides of the equation by 4ac:
b^2/(4ac) = 1
Simplify the equation:
ab^2 = 4c
Now, we need to examine the equation ab^2 = 4c to find the possible values of ab.
Since a and c can be any real numbers, let's consider the extreme cases:
Assume a = c = 1, then the equation becomes:
b^2 = 4
b = ±2
Assume a = c = -1, then the equation becomes:
b^2 = -4
b = ±2i (complex roots)
From these calculations, we can see that ab can take any value in the range (-∞, -2] ∪ [2, +∞).
Now, let's analyze the answer choices:
(A) ab >= 1/8
(B) ab <= 1/8
(C) ab >= 1/4
(D) ab <= 1/4
From our earlier calculation, we know that ab can take any value in the range (-∞, -2] ∪ [2, +∞). None of the answer choices are within these ranges.
Therefore, none of the answer choices (A), (B), (C), or (D) are true.
Hence, the correct answer is none of the above options.