Question
Solve:
$4 x^2-12 x+5$
$4 x^2-12 x+5$
✓
Answer
$4 x^2-12 x+5$
$=\Big(\text{x}^2 - 3\text{x} +\frac{5}{4}\Big)$ [Making the co-efficient of $\left.x^2=1\right]$
$=4\Big[\text{x}^2-3\text{x}+\Big(\frac{3}{2}\Big)^2-\Big(\frac{3}{2}\Big)^2+ \frac{5}{4}\Big]$ $\Big[$Adding and subtracting $\Big(\frac{3}{2}\Big)^2\Big]$
$=4\Big[\Big(\text{x}-\frac{3}{2}\Big)^2-\frac{9}{4}+\frac{5}{4}\Big]$ [Compliting the suare]
$=4\Big[\Big(\text{x}-\frac{3}{2}\Big)^2 -1\Big]$
$=4\Big[\Big(\text{x}-\frac{3}{2}\Big)-1\Big]\Big[\Big(\text{x}-\frac{3}{2}\Big)+1\Big]$
$=4\Big(\text{x}-\frac{3}{2}-1\Big)\Big(\text{x}-\frac{3}{2}+1\Big)$
$=4\Big(\text{x}-\frac{5}{2}\Big)\Big(\text{x}-\frac{1}{2}\Big)$
$=(2\text{x}-5)(2\text{x}-1)$
$=\Big(\text{x}^2 - 3\text{x} +\frac{5}{4}\Big)$ [Making the co-efficient of $\left.x^2=1\right]$
$=4\Big[\text{x}^2-3\text{x}+\Big(\frac{3}{2}\Big)^2-\Big(\frac{3}{2}\Big)^2+ \frac{5}{4}\Big]$ $\Big[$Adding and subtracting $\Big(\frac{3}{2}\Big)^2\Big]$
$=4\Big[\Big(\text{x}-\frac{3}{2}\Big)^2-\frac{9}{4}+\frac{5}{4}\Big]$ [Compliting the suare]
$=4\Big[\Big(\text{x}-\frac{3}{2}\Big)^2 -1\Big]$
$=4\Big[\Big(\text{x}-\frac{3}{2}\Big)-1\Big]\Big[\Big(\text{x}-\frac{3}{2}\Big)+1\Big]$
$=4\Big(\text{x}-\frac{3}{2}-1\Big)\Big(\text{x}-\frac{3}{2}+1\Big)$
$=4\Big(\text{x}-\frac{5}{2}\Big)\Big(\text{x}-\frac{1}{2}\Big)$
$=(2\text{x}-5)(2\text{x}-1)$
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