Question
Solve the following equations by using the method of completing the square:
$8x^2 - 14x - 15 = 0$
$8x^2 - 14x - 15 = 0$
✓
Answer
$8x^2 - 14x - 15 = 0$
$\Rightarrow 16x^2 - 28x - 30 = 0$ (Multiplying both sides by 2)
$\Rightarrow 16x^2 - 28x = 30$
$\Rightarrow(\text{4x})^2-2\times\text{4x}\times\frac{7}{2}+\Big(\frac{7}{2}\Big)^2\\=30+\Big(\frac{7}{2}\Big)^2$ $\Big[$Adding $\Big(\frac{7}2{}\Big)^2$ on both sides$\Big]$
$\Rightarrow\Big(\text{4x}-\frac{7}{2}\Big)^2$
$=30+\frac{49}{4}$
$=\frac{169}{4}=\Big(\frac{13}{2}\Big)^2$
$\Rightarrow\text{4x}-\frac{7}{2}=\pm\frac{13}{2}$ (Taking square root on both sides)
$\Rightarrow\text{4x}-\frac{7}{2}=\frac{13}{2}$ or $\text{4x}-\frac{7}{2}=-\frac{13}{2}$
$\Rightarrow\text{4x}=\frac{13}{2}+\frac{7}{2}=\frac{20}{2}=10$ or $\text{4x}=-\frac{13}{2}+\frac{7}{2}=-\frac{6}{2}=-3$
$\Rightarrow\text{x}=\frac{5}{2}$ or $\text{x}=-\frac{3}4{}$
Hence, $\frac{5}{2}$ and $-\frac{3}{4}$ are the roots of the given equation.
$\Rightarrow 16x^2 - 28x - 30 = 0$ (Multiplying both sides by 2)
$\Rightarrow 16x^2 - 28x = 30$
$\Rightarrow(\text{4x})^2-2\times\text{4x}\times\frac{7}{2}+\Big(\frac{7}{2}\Big)^2\\=30+\Big(\frac{7}{2}\Big)^2$ $\Big[$Adding $\Big(\frac{7}2{}\Big)^2$ on both sides$\Big]$
$\Rightarrow\Big(\text{4x}-\frac{7}{2}\Big)^2$
$=30+\frac{49}{4}$
$=\frac{169}{4}=\Big(\frac{13}{2}\Big)^2$
$\Rightarrow\text{4x}-\frac{7}{2}=\pm\frac{13}{2}$ (Taking square root on both sides)
$\Rightarrow\text{4x}-\frac{7}{2}=\frac{13}{2}$ or $\text{4x}-\frac{7}{2}=-\frac{13}{2}$
$\Rightarrow\text{4x}=\frac{13}{2}+\frac{7}{2}=\frac{20}{2}=10$ or $\text{4x}=-\frac{13}{2}+\frac{7}{2}=-\frac{6}{2}=-3$
$\Rightarrow\text{x}=\frac{5}{2}$ or $\text{x}=-\frac{3}4{}$
Hence, $\frac{5}{2}$ and $-\frac{3}{4}$ are the roots of the given equation.
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