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Solving Equations with Linear Expressions on One Side and Numbers on the Other Side

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Question

Solve each of the following equations and also verify your solution :

(x + 2)(x+3)+(x-3)(x-2)-2x(x+1)=0

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Solution

$(x + 2) (x + 3) + (x - 3) (x - 2) - 2 x (x + 1) = 0 \Rightarrow [x^{2} + (2 + 3) x + 2 \times 3] + [x^{2} + (- 3 - 2) x + (- 3) (- 2)] - 2 x^{2} - 2 x = 0 \Rightarrow x^{2} + 5 x + 6 + x^{2} - 5 x + 6 - 2 x^{2} - 2 x = 0 \Rightarrow x^{2} + x^{2} - 2 x^{2} + 5 x - 5 x - 2 x + 6 + 6 = 0 = - 2 x + 12 = 0$
Substracting 12 from both sides,
$- 2 x + 12 - 12 = 0 - 12 \Rightarrow - 2 x = - 12$
Dividing by - 2,
$\frac{- 2 x}{- 2} = \frac{- 12}{- 2} \Rightarrow x = 6 ∴ x = 6$
Verification :
$L . H . S . = (x + 2) (x + 3) + (x - 3) (x - 2) - 2 x (x + 1) = (6 + 2) (6 + 3) + (6 - 3) (6 - 2) - 2 \times 6 (6 + 1) = 8 \times 9 + 3 \times 4 - 12 \times 7 = 72 + 12 - 84 = 84 - 84 = 0 = R . H . S .$

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