1) \( x^{2}=2 x+3 \)

To solve the equation \( x^{2} = 2x + 3 \), you can rearrange it into standard quadratic form by moving all terms to one side. This gives you \( x^{2} - 2x - 3 = 0 \). This equation can be factored as \( (x - 3)(x + 1) = 0 \), leading to the solutions \( x = 3 \) and \( x = -1 \). These solutions can be quickly verified by substituting them back into the original equation, ensuring both satisfy \( x^{2} = 2x + 3 \). Quadratic equations frequently pop up in various contexts, such as physics when calculating projectile motion or in optimization problems in economics!