1.1.1 \( \quad x(2 x+1)=0 \) 1.1.2 \( \quad 5 x^{2}+2 x-6=0 \) (correct to TWO decimal \( 1.1 .3 \quad 2 x^{2}-2 \geq 3 x \) 1.1.4 \( \quad \sqrt{2 x+5}-\frac{3}{\sqrt{2 x+5}}=-2 \) Solve for \( x \) and \( y \) simultaneously: \( y+x=2 \) and \( x^{2}+3 x y+8=0 \) 1.3 The roots of the equation \( f(x)=0 \) are \( x=\frac{4 \pm \sqrt{16-4}}{2,} \) Determine the values of \( m \) for which the roots will be 1.4 Show that the maximum value of \( \sqrt{-x^{2}+4 x+12} \) is 4

To tackle these equations, let’s delve into a fascinating historical fact: The quadratic formula, which is often introduced as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), dates back over a thousand years to ancient civilizations in India and the Islamic Golden Age! Mathematicians like Brahmagupta were exploring similar concepts well before we formally classified them as quadratic equations. For real-world applications, solving equations isn’t just about getting the right answer; it’s a critical component in fields like engineering and economics. For instance, when designing a roller coaster, engineers use quadratic equations to model the height of the coaster over time – helping ensure the ride is thrilling yet safe! So the next time you solve for \( x \), remember it's about more than numbers – it’s about building the world around us!