Question

If the sum of the roots of the equation ax² + 4x + c = 0 is half of their difference , then the value of ac is ? ♣

Answer 1

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Answer 2

Concept-

Use the concept of sum of roots of quadratic equation . Sum of roots of quadratic equation is equal to the negation of coefficient of second term, divided by leading coefficient.

Given-

Sum of the roots of the equation ax² + 4x + c = 0 is half of their difference.

Find-

Find the value of ac.

Solution-

ax² + 4x + c = 0

According to the question,

∝ + β = ∝ - β / 2    [ Here, ∝ and β are roots.]

∵ ∝ + β = -4/a    →   equation (1)

   ∝β = c/a         →  equation (2)

Since, squaring both sides , we get

(∝ + β)² = { (∝ - β) / 2 }²

4 (∝ + β)² = (∝ - β)²

4 (∝ + β)² = (∝² + β² - 2∝β + 4∝β - 4∝β)

4 (∝ + β)² = (∝ + β)² - 4∝β

4 (∝ + β)² - (∝ + β)² = -4∝β

3 (∝ + β)² = -4∝β

3 (-4/a)² = -4 × c/a

48/a² = -4c/a

ac = -12

a = 1 , c = -12

⇒ x² + 4x - 12 = 0

⇒ roots are -6 and 2

sum = -4

difference =-8

Hence, the value of ac is -12.

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