Question

Solve the following quadratic equation:

9x² – 9(a + b)x + (2a² + 5ab + 2b²) = 0

Answer 1

We write, –9(a + b)x = –3(2a + b)x – 3(a + 2b)x as 

9x² × (2a² + 5ab + 2b²) = 9(2a² + 5ab + 2b²)x² = [–3(2a + b)x] × [–3(a + 2b)x] 

∴ 9x² – 9(a + b)x + (2a² + 5ab + 2b²) = 0 

⇒ 9x² – 3(2a + b)x – 3(a + 2b)x + (2a + b)(a + 2b) = 0 

⇒ 3x[3x – (2a + b)] – (a + 2b)[3x – (2a + b)] = 0 

⇒ [3x – (2a + b)][3x – (a + 2b)] = 0 

⇒ 3x – (2a + b) = 0 or 3x – (a + 2b) = 0 

⇒ `x = (2a + b)/3` or `x = (a + 2b)/3` 

Hence, `(2a + b)/3` and `(a + 2b)/3` are the roots of the given equation.