Question

Solve the given quadratic equation for x : 9x² – 9(a + b)x + (2a² + 5ab + 2b²) = 0 ?

Answer 1

\[9 x^2 - 9\left( a + b \right)x + \left( 2 a^2 + 5ab + 2 b^2 \right) = 0\]
\[ \Rightarrow 9 x^2 - 3\left\{ \left( 2a + b \right) + \left( a + 2b \right) \right\} + \left( 2 a^2 + 4ab + ab + 2 b^2 \right) = 0\]
\[ \Rightarrow 9 x^2 - 3\left\{ \left( 2a + b \right) + \left( a + 2b \right) \right\}x + \left\{ 2a\left( a + 2b \right) + b\left( a + 2b \right) \right\} = 0\]
\[ \Rightarrow 9 x^2 - 3\left\{ \left( 2a + b \right) + \left( a + 2b \right) \right\}x + \left( 2a + b \right)\left( a + 2b \right) = 0\]

\[\Rightarrow 9 x^2 - 3\left( 2a + b \right)x - 3\left( a + 2b \right)x + \left( 2a + b \right)\left( a + 2b \right) = 0\]
\[ \Rightarrow 3x\left\{ 3x - \left( 2a + b \right) \right\} - \left( a + 2b \right)\left\{ 3x - \left( 2a + b \right) \right\} = 0\]
\[ \Rightarrow \left\{ 3x - \left( a + 2b \right) \right\}\left\{ 3x - \left( 2a + b \right) \right\} = 0\]
\[ \Rightarrow 3x - \left( a + 2b \right) = 0 or 3x - \left( 2a + b \right) = 0\]
\[ \Rightarrow 3x = a + 2b or 3x = 2a + b\]
\[ \Rightarrow x = \frac{a + 2b}{3} or x = \frac{2a + b}{3}\]