Given x2+bx+c=0 let d be positive difference in the roots if b c d form increasing sequence of consecutive integers then sum b+c+d is

Question

Given x2+bx+c=0 let d be the positive difference in the roots if b c and  d form an increasing sequence of consecutive integers then sum of b+c+d=

Answer 1

So a = 1

Having b, c, & d be consecutive integers tells us that:

b = c - 1

c = c

d = c + 1

Knowing your quadratic formula helps. To find the roots of a parabola, we solve for x where y = 0, so

x = [-b±√(b² - 4ac)]/(2a) = -b/(2a) ± √(b² - 4ac)/(2a)

Those are the two roots for any generic parabola. So we subtract them and take the absolute value to find the distance between them:

|[-b/(2a) + √(b² - 4ac)/(2a)] - [-b/(2a) - √(b² - 4ac)/(2a)]|

 = |-b/(2a) + √(b² - 4ac)/(2a) + b/(2a) + √(b² - 4ac)/(2a)|

 = |2√(b² - 4ac)/(2a)|

 = √(b² - 4ac)/a = d

This is true for any parabola. Looking back at where I bolded the quadratic function,

• the -b/(2a) is actually the x-value of our vertex, right in the middle of the x's of the two roots
• √(b² - 4ac)/a is always the distance between the two roots because the ± means that, from the vertex x-value, you'll go to the right (+) half of that & you'll go to the left (-) half of that to solve for the roots

Moving on.....with this equation, we can plug in for a, b & d and finally solve for c:

√(b² - 4ac)/a = d

√((c-1)² - 4×1×c)/1 = c + 1

√(c² - 2c + 1 - 4c) = c + 1

√(c² - 6c + 1) = c + 1

c² - 6c + 1 = (c + 1)²

c² - 6c + 1 = c² + 2c + 1

8c = 0

c = 0

So your consecutive integers must be -1, 0, 1. And, therefore, your sum must be 0.

Have a good weekend!