Tìm x biết: (x + 5)² - (x - 5)² - 2x + 1 = 0

Lời giải:

G = x² – x + 2y² – 4y + 3

G = \({\left( {x - \frac{1}{2}} \right)^2} + 2{\left( {y - 1} \right)^2} + \frac{3}{4}\)

Vì \({\left( {x - \frac{1}{2}} \right)^2} + 2{\left( {y - 1} \right)^2} \ge 0,\forall x,y \Rightarrow {\left( {x - \frac{1}{2}} \right)^2} + 2{\left( {y - 1} \right)^2} + \frac{3}{4} \ge \frac{3}{4},\forall x,y\)

Vậy GTNN của G là \(\frac{3}{4}\) khi \(\left\{ \begin{array}{l}x = \frac{1}{2}\\y = 1\end{array} \right.\)

Lời giải:

(3x – 7)² – 4(x + 1)² = 0

⇔ (3x – 7)² – [2(x + 1)]² = 0

⇔ (3x – 7)² – (2x + 2)² = 0

⇔ (3x – 7 + 2x + 2)(3x – 7 – 2x – 2) = 0

⇔ (5x – 5)(x – 9) = 0

⇔ \(\left[ \begin{array}{l}x = 1\\x = 9\end{array} \right.\)

Vậy x = 1 hoặc x = 9.