If one root of the equationax2+bx+c=0is three times the other, thenb2:ac=a)3 : 1b)16 : 3c)3 : 16d)16 : 1Correct answer is option 'B'. Can you explain this answer?

Question

Answer

To solve this problem, let's assume that the roots of the quadratic equation ax^2 + bx + c = 0 are x and 3x, where x is a constant.

Finding the sum and product of the roots:
The sum of the roots is given by the formula -b/a, and the product of the roots is given by the formula c/a.

So, the sum of the roots (x + 3x) is -b/a, which simplifies to 4x = -b/a.

Similarly, the product of the roots (x * 3x) is c/a, which simplifies to 3x^2 = c/a.

Finding the ratio of b^2 to ac:
We need to find the ratio b^2 : ac.

Substituting the value of 4x for -b/a in the equation 4x = -b/a, we get:
b = -4ax.

Now, substituting the value of 3x^2 for c/a in the equation 3x^2 = c/a, we get:
c = 3ax^2.

Substituting these values of b and c into the ratio b^2 : ac, we get:
(-4ax)^2 : (3ax^2)(a)
(16a^2x^2) : (3a^2x^2)
16 : 3.

Therefore, the ratio of b^2 to ac is 16 : 3.

Thus, the correct answer is option B) 16 : 3.

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