Question
Solve the following quadratic equations by factorization:
$ a^2 x^2-3 a b x+2 b^2=0 $
$ a^2 x^2-3 a b x+2 b^2=0 $
✓
Answer
We have,
$ a^2 x^2-3 a b x+2 b^2=0 $
$ \Rightarrow a^2 x^2-a b x-2 a b x+2 b^2=0 $
$ {\left[a^2 \times 2 b^2=2 a^2 b^2 \Rightarrow 2 a^2 b^2=2 a b \times a b=-2 a b \times-a b \Rightarrow-3 a b=-2 a b-a b\right]}$
$⇒ ax(ax - b) - 2b(ax - b) = 0$
$⇒ (ax - 2b)(ax - b) = 0$
$⇒ ax = 2b$ or $ax = b$
$\Rightarrow\text{x}=\frac{2\text{b}}{\text{a}}$ or $\text{x}=\frac{\text{b}}{\text{a}}$
$\therefore\text{x}=\frac{\text{b}}{\text{a}}$ and $\text{x}=\frac{2\text{b}}{\text{a}}$ are the two roots of the given quadratic equation.
$ a^2 x^2-3 a b x+2 b^2=0 $
$ \Rightarrow a^2 x^2-a b x-2 a b x+2 b^2=0 $
$ {\left[a^2 \times 2 b^2=2 a^2 b^2 \Rightarrow 2 a^2 b^2=2 a b \times a b=-2 a b \times-a b \Rightarrow-3 a b=-2 a b-a b\right]}$
$⇒ ax(ax - b) - 2b(ax - b) = 0$
$⇒ (ax - 2b)(ax - b) = 0$
$⇒ ax = 2b$ or $ax = b$
$\Rightarrow\text{x}=\frac{2\text{b}}{\text{a}}$ or $\text{x}=\frac{\text{b}}{\text{a}}$
$\therefore\text{x}=\frac{\text{b}}{\text{a}}$ and $\text{x}=\frac{2\text{b}}{\text{a}}$ are the two roots of the given quadratic equation.
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