Solve for $x$ if $| x^{2} - 4 x + 3 | = | x^{2} - 2 x + 1 |$

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Solution

$\begin{matrix} ∣ ∣ x^{2} - 4 x + 3 ∣ ∣ = ∣ ∣ x^{2} - 2 x + 1 ∣ ∣ w e h a v e 4 c a s e (i) x^{2} - 4 x + 3 = x^{2} - 2 x + 1 3 - 1 = 9 x - 2 x 2 = 2 x x = 1 (i i) - (x^{2} - 4 x + 3) = x^{2} - 2 x + 1 - x^{2} + 4 x - 3 = x^{2} - 2 x + 1 = 2 x^{2} - 6 + 4 = 0 \Rightarrow 2 x^{2} - 4 x - 2 x + 4 2 x (x - 2) - 2 (x - 2) = 0 (2 x - 2) (x - 2) x = 1, x = 2 (i i i) (x^{2} - 4 x + 3) = - (x^{2} - 2 x + 1) x^{2} - 4 x + 3 = - x^{2} + 2 x - 1 2 x^{2} - 6 x + 4 = 0 x = 1, 2 \end{matrix}$

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