1.

For 4x2 - 18x + 20 = 0, find a3 + b3 where a and b are the roots of the equation.1. 20.6252. 12.6253. 23.6254. Insufficient Data

Answer» Correct Answer - Option 3 : 23.625

Concept:

For an equation ax2 + bx +c = 0

  • Sum of the roots = \(\rm-b\over a\)
  • Product of the roots = \(\rm c\over a\)
  • Roots of the equation = \(\rm x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
  • If α and β are the roots of the equation the equation can be represented as (x - α)(x - β) = 0

The roots of a quadratic equation ax2 +bx + c = 0 are real if:

b2 - 4ac ≥ 0

Calculation:

For a and b are the roots of 4x2 - 18x + 20 = 0 then, 

a + b = \(\rm 18\over 4\) = 4.5 

ab = \(\rm 20\over 4\) = 5

(a + b)2 = a2 + b2 + 2ab

⇒ 4.5 × 4.5 = a2 + b2 + 2(5)

⇒ a2 + b2  = 20.25 - 10

⇒ a2 + b2  = 10.25

Now a3 + b3  = (a + b)(a2 + b2 - ab)

⇒ a3 + b3  = (4.5)(10.25 - 5)

⇒ a3 + b3  = (4.5) × (5.25)

⇒ a3 + b3   = 23.625


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