(i) \( 3 x^{2}-4 x=-4 x+5 \)

To solve the equation \( 3x^{2} - 4x = -4x + 5 \), first, we can simplify it by combining like terms. Adding \( 4x \) to both sides gives us \( 3x^{2} - 4x + 4x = 5 \), which simplifies to \( 3x^{2} = 5 \). Next, divide both sides by 3 to isolate \( x^{2} \): \( x^{2} = \frac{5}{3} \). Now, taking the square root of both sides gives us \( x = \pm \sqrt{\frac{5}{3}} \) or simplifying further \( x = \pm \frac{\sqrt{15}}{3} \). For a quick check on your solution, plug the values of \( x = \frac{\sqrt{15}}{3} \) and \( x = -\frac{\sqrt{15}}{3} \) back into the original equation. This not only verifies your solutions but also helps solidify your understanding of why they work. It’s a great way to catch any potential mistakes along the way!