Question
Solve the following quadratic equation:
$9x^2 - 9(a + b)x + (2a^2 + 5ab + 2b^2) = 0$
$9x^2 - 9(a + b)x + (2a^2 + 5ab + 2b^2) = 0$
✓
Answer
$9x^2 - 9(a + b)x + (2a^2 + 5ab + 2b^2)$
$\Rightarrow 9x^2 - 9(a + b)x + (2a^2 + 4ab + ab + 2b^2) = 0$
$\Rightarrow 9x^2 - 9(a + b)x + [2a(a + 2b) + b(a + 2b)] = 0$
$\Rightarrow 9x^2 - 9(a + b)x + (a + 2b)(2a + b) = 0$
$\Rightarrow 9x^2 - 3(a + 2b)x - 3(2a + b)x + (a + 2b)(2a + b) = 0$
$\Rightarrow 3x[3x - (a + 2b)] - (2a + b)[3x - (a + 2b)] = 0$
$\Rightarrow [3x - (a + 2b)][3x - (2a + b)] = 0$
$\Rightarrow 3x - (a + 2b) = 0$ or $3x - (2a + b) = 0$
$\Rightarrow\text{x}=\frac{\text{a}+\text{2b}}{3}$ or $\text{x}=\frac{\text{2a}+\text{b}}{3}$
$\Rightarrow 9x^2 - 9(a + b)x + (2a^2 + 4ab + ab + 2b^2) = 0$
$\Rightarrow 9x^2 - 9(a + b)x + [2a(a + 2b) + b(a + 2b)] = 0$
$\Rightarrow 9x^2 - 9(a + b)x + (a + 2b)(2a + b) = 0$
$\Rightarrow 9x^2 - 3(a + 2b)x - 3(2a + b)x + (a + 2b)(2a + b) = 0$
$\Rightarrow 3x[3x - (a + 2b)] - (2a + b)[3x - (a + 2b)] = 0$
$\Rightarrow [3x - (a + 2b)][3x - (2a + b)] = 0$
$\Rightarrow 3x - (a + 2b) = 0$ or $3x - (2a + b) = 0$
$\Rightarrow\text{x}=\frac{\text{a}+\text{2b}}{3}$ or $\text{x}=\frac{\text{2a}+\text{b}}{3}$
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