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r and s are the roots of the quadratic equation ax2 + bx + c = 0 where a ≠ 0 & s >0, such that r is 50 percent greater than s. If the product of the roots of the equation is 150, what is the sum of the roots of the equation?
  • a)
    -25
  • b)
    -15
  • c)
    15
  • d)
    25
  • e)
    Cannot be determined
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
r and s are the roots of the quadratic equation ax2 + bx + c = 0 where...
Given         
  • ax2+bx+c=0
    • a ≠ 0
    • s>0
    • Roots of the equation are (r, s)
    • r = s + 50% of s = 1.5s
  • r * s = 150
To Find:  r + s?
Approach
  1. For finding the values of r + s, we need to find the values of r and s
  2. We are given two relations between r and s, which are:
       a.  r = 1.5s and
       b.  r * s = 150
  3. As we have 2 equations and 2 variables, we can find the values of r and s and hence the value of r + s.
 
Working Out
  1. Finding values of r and s
r=1.5s……..(1)
r*s = 150……..(2)
Substituting r = 1.5s in (2), we have
1.5s2 = 150
s2 = 100, i.e. s = 10 or -10
  1. If s = 10, r = 15
         a. So, r + s = 10 + 15 = 25
  2. If s = -10,
    1. However, we are given s >0
    2. Hence, we can reject this case.
  3. So, the value of r + s = 25
Hence the sum of the roots of the equation ax2+bx+c=0
 is 25
 
Answer: D
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Most Upvoted Answer
r and s are the roots of the quadratic equation ax2 + bx + c = 0 where...
To solve this problem, let's start by assigning variables to the roots of the quadratic equation. Let's say that r is one root and s is the other root.

Given that r is 50% greater than s, we can express this relationship as:

r = 1.5s

We are also given that the product of the roots is 150. This means that:

rs = 150

Now, let's express the quadratic equation in terms of the roots:

ax^2 + bx + c = 0

Since r and s are the roots, we can write the equation as:

a(x - r)(x - s) = 0

Expanding this equation, we get:

ax^2 - (ar + as)x + rs = 0

Now, let's substitute the values of r and s into the equation:

ax^2 - (1.5as + as)x + 150 = 0

Simplifying further, we get:

ax^2 - 2.5asx + 150 = 0

Since the quadratic equation is in the form ax^2 + bx + c = 0, we can equate the coefficients to find the sum of the roots:

b/a = -2.5as/a

b = -2.5as

Since the sum of the roots is equal to -b/a, we can substitute the value of b:

Sum of roots = -(-2.5as)/a

Sum of roots = 2.5s

Now, we can substitute the value of rs = 150 into the equation rs = 150:

s(1.5s) = 150

1.5s^2 = 150

Dividing both sides by 1.5, we get:

s^2 = 100

Taking the square root of both sides, we get:

s = 10

Now, substituting the value of s into the equation for the sum of the roots:

Sum of roots = 2.5s

Sum of roots = 2.5(10)

Sum of roots = 25

Therefore, the sum of the roots of the quadratic equation is 25. So, the correct answer is option D) 25.
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Marginal analysis is an important decision-making tool in the business world. Pricing decisions tend to heavily involve analysis regarding marginal contributions to revenues and costs. In business, the practice of setting the price of a product to equal the extra cost of producing an extra unit of output, i.e. the marginal cost of producing the unit, is known as marginal-cost pricing. In the marginal analysis of pricing decisions, if marginal revenue, the increase in revenue from the sale of an additional unit of output, is greater than marginal cost at some level of output, marginal profit is positive, and, therefore, a greater quantity should be produced. Alternatively, if marginal revenue is less than the marginal cost, marginal profit is negative and a lesser quantity should be produced. Accordingly, firms tend to use this analysis to increase their production until marginal revenue equals marginal cost, and then charge a price which is determined by the demand curve. For instance, businesses often set prices close to marginal cost during periods of poor sales. If, for example, an item has a marginal cost of $1.00 and a normal selling price of $2.00, the firm selling the item might wish to lower the price to $1.10 - if demand has waned. The business would choose this approach because the incremental profit of 10 cents from the transaction is better than no sale at all.Which of the following is supported by the information given in the passage?

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r and s are the roots of the quadratic equation ax2 + bx + c = 0 where a ≠ 0 & s >0, such that r is 50 percent greater than s. If the product of the roots of the equation is 150, what is the sum of the roots of the equation?a)-25b)-15c)15d)25e)Cannot be determinedCorrect answer is option 'D'. Can you explain this answer?
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