Question
Solve the following equations by using the method of completing the square:
$5 x^2-6 x-2=0$
$5 x^2-6 x-2=0$
✓
Answer
$5 x^2-6 x-2=0$
$\left.\Rightarrow 25 x^2-30 x-10=0 \text { (Multiplying both sides by } 5\right)$
$\Rightarrow 25 x^2-30 x=10$
$\Rightarrow(5 x)^2-2 \times 5 x \times 3+3^2=10+3^2 \text { [Adding } 3^2 \text { on both sides] }$
$\Rightarrow(5 x-3)^2=10+9=19$
$\Rightarrow\text{5x}-3=\pm19$ (Taking square root on both sides)
$\Rightarrow\text{5x}-3=\sqrt{19}$ or $\text{5x}- 3=-\sqrt{19}$
$\Rightarrow\text{5x}=3+\sqrt{19}$ or $\text{5x}=3-\sqrt{19}$
$\Rightarrow\text{x}=\frac{3+\sqrt{19}}{5}$ or $\text{x}=\frac{3-\sqrt{19}}{5}$
Hence, $\frac{3+\sqrt{19}}{5}$ and $\frac{3-\sqrt{19}}{5}$ are the roots of the given equation.
$\left.\Rightarrow 25 x^2-30 x-10=0 \text { (Multiplying both sides by } 5\right)$
$\Rightarrow 25 x^2-30 x=10$
$\Rightarrow(5 x)^2-2 \times 5 x \times 3+3^2=10+3^2 \text { [Adding } 3^2 \text { on both sides] }$
$\Rightarrow(5 x-3)^2=10+9=19$
$\Rightarrow\text{5x}-3=\pm19$ (Taking square root on both sides)
$\Rightarrow\text{5x}-3=\sqrt{19}$ or $\text{5x}- 3=-\sqrt{19}$
$\Rightarrow\text{5x}=3+\sqrt{19}$ or $\text{5x}=3-\sqrt{19}$
$\Rightarrow\text{x}=\frac{3+\sqrt{19}}{5}$ or $\text{x}=\frac{3-\sqrt{19}}{5}$
Hence, $\frac{3+\sqrt{19}}{5}$ and $\frac{3-\sqrt{19}}{5}$ are the roots of the given equation.
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