Question

If the equation 8x² - 6x + a = 0 has one root equal to the square of the other root, then find the value of a.
1. -27 or 1
2. -24 or 1
3. -25 or 2
4. 0 or 2

Answer 1

Correct Answer - Option 1 : -27 or 1

Concept:

For an quadratic equation ax² + bx + c = 0, 

Sum of roots = \(\rm -b\over a\)

Product of roots = \(\rm c\over a\) 

 

Calculation:

Let the one root be α, the other root be α² 

Given equation 8x² - 6x + a = 0

Sum of roots = \(-{-6\over8}\)

α + α2 = \(3\over4\)          ...(i)

Product of the roots = \(\rm a\over8\)

α × α² = \(\rm a\over8\) 

α³ = \(\rm a\over8\)             ...(ii)

Adding (i) and (ii), we get

α + α2 + α3 = \(\rm {3\over4}+{a\over8}\)

α (1 + α + α2) = \(\rm 6+a\over8\)

From equation (i)

α (1 + \(\rm 3\over4\)) = \(\rm 6+a\over8\)

α (\(\rm 7\over4\)) = \(\rm 6+a\over8\)

α = \(\rm 6+a\over14\)

Putting in equation (i)

\(\rm 6+a\over14\) + \(\rm \left(6+a\over14\right)^2\) = \(3\over4\)

\(\rm14a+84+a^2+12a+36\over49\) = 3

a² + 26a + 120 = 147

a² + 26a - 27 = 0

(a + 27)(a - 1) = 0

a = -27 or 1