Question
Solve the following equations by using the method of completing the square:
$2x^2 + 5x - 3 = 0$
$2x^2 + 5x - 3 = 0$
✓
Answer
$2x^2 + 5x - 3 = 0$
$\Rightarrow 4x^2 + 10x - 6 = 0$ (Multiplying both sides by $2$)
$\Rightarrow 4x^2 + 10x = 6$
$\Rightarrow(\text{2x})^2+2\times\text{2x}\times\frac{5}{2}+\Big(\frac{5}{2}\Big)^2\\=6+\Big(\frac{5}{2}\Big)^2$ $\Big[$Adding $\Big(\frac{5}2{}\Big)^2$ on both sides$\Big]$
$\Rightarrow\Big(\text{2x}+\frac{5}{2}\Big)^2$
$=6+\frac{25}{4}$
$=\frac{24+25}{4}=\frac{49}{7}=\Big(\frac{7}{2}\Big)^2$
$\Rightarrow\text{2x}+\frac{5}{2}=\pm\frac{7}{2}$ (Taking square root on both sides)
$\Rightarrow\text{2x}+\frac{5}{2}=\frac{7}{2}$ or $\text{2x}+\frac{5}{2}=-\frac{7}{2}$
$\Rightarrow\text{2x}=\frac{7}{2}-\frac{5}{2}=\frac{2}{2}=1$ or $\text{2x}=-\frac{7}{2}-\frac{5}{2}=-\frac{12}{2}=-6$
$\Rightarrow\text{x}=\frac{1}{2}$ or $x = -3$
Hence $\frac{1}{2}$ and -3 are the roots of the given equation.
$\Rightarrow 4x^2 + 10x - 6 = 0$ (Multiplying both sides by $2$)
$\Rightarrow 4x^2 + 10x = 6$
$\Rightarrow(\text{2x})^2+2\times\text{2x}\times\frac{5}{2}+\Big(\frac{5}{2}\Big)^2\\=6+\Big(\frac{5}{2}\Big)^2$ $\Big[$Adding $\Big(\frac{5}2{}\Big)^2$ on both sides$\Big]$
$\Rightarrow\Big(\text{2x}+\frac{5}{2}\Big)^2$
$=6+\frac{25}{4}$
$=\frac{24+25}{4}=\frac{49}{7}=\Big(\frac{7}{2}\Big)^2$
$\Rightarrow\text{2x}+\frac{5}{2}=\pm\frac{7}{2}$ (Taking square root on both sides)
$\Rightarrow\text{2x}+\frac{5}{2}=\frac{7}{2}$ or $\text{2x}+\frac{5}{2}=-\frac{7}{2}$
$\Rightarrow\text{2x}=\frac{7}{2}-\frac{5}{2}=\frac{2}{2}=1$ or $\text{2x}=-\frac{7}{2}-\frac{5}{2}=-\frac{12}{2}=-6$
$\Rightarrow\text{x}=\frac{1}{2}$ or $x = -3$
Hence $\frac{1}{2}$ and -3 are the roots of the given equation.
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