Question
Solve the following equations by using the method of completing the square:
$4x^2 + 4bx - (a^2 - b^2) = 0$
$4x^2 + 4bx - (a^2 - b^2) = 0$
✓
Answer
$4x^2 + 4bx - (a^2 - b^2) = 0$
$\Rightarrow 4x^2 + 4bx = a^2 - b^2$
$\Rightarrow (2x)^2 + 2 \times 2x \times b + b^2 = a^2- b^2 + b^2$ [Adding $b^2$ on both sides]
$\Rightarrow (2x + b)^2 = a^2$
$\Rightarrow\text{2x}+\text{b}=\pm\text{a}$ (Taking square root on both sides)
$\Rightarrow 2x + b = a$ or $2x + b = -a$
$\Rightarrow 2x = a - b$ or $2x = -a - b$
$\Rightarrow\text{x}=\frac{\text{a}-\text{b}}{2}$ or $\text{x}=-\frac{\text{a}+\text{b}}{2}$
Hence, $\frac{\text{a}-\text{b}}{2}$ and $-\frac{\text{a}+\text{b}}{2}$ are the roots of the given equation.
$\Rightarrow 4x^2 + 4bx = a^2 - b^2$
$\Rightarrow (2x)^2 + 2 \times 2x \times b + b^2 = a^2- b^2 + b^2$ [Adding $b^2$ on both sides]
$\Rightarrow (2x + b)^2 = a^2$
$\Rightarrow\text{2x}+\text{b}=\pm\text{a}$ (Taking square root on both sides)
$\Rightarrow 2x + b = a$ or $2x + b = -a$
$\Rightarrow 2x = a - b$ or $2x = -a - b$
$\Rightarrow\text{x}=\frac{\text{a}-\text{b}}{2}$ or $\text{x}=-\frac{\text{a}+\text{b}}{2}$
Hence, $\frac{\text{a}-\text{b}}{2}$ and $-\frac{\text{a}+\text{b}}{2}$ are the roots of the given equation.
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