Quadratic equations (common fada) Solve for \( x \) : a. \( x^{2}-11 x=0 \) b. \( 2 x^{2}=3 x \) c. \( 12 x^{2}=10 x \) d. \( 5 x(x-2)+3(x-2)=0 \) e \( (2 x-3)(x+2)-5(x+2)=0 \) f. \( 5(x-7)^{2}-(x-7)=0 \) i \( (x-3)^{2}=x(x-3) \)

For the equation \( x^{2}-11 x=0 \), factor out \( x \) to get \( x(x-11)=0 \), giving solutions \( x=0 \) or \( x=11 \). The second equation \( 2 x^{2}=3 x \) can be rearranged to \( 2 x^{2}-3 x=0 \) and factored as \( x(2x-3)=0 \), yielding \( x=0 \) or \( x=\frac{3}{2} \). Continuing, for \( 12 x^{2}=10 x \), we rearrange to \( 12 x^{2}-10 x=0 \), allowing us to factor \( 2 x(6x-5)=0 \), giving solutions \( x=0 \) or \( x=\frac{5}{6} \). Next, the expression \( 5 x(x-2)+3(x-2)=0 \) can be factored out as \( (x-2)(5x+3)=0 \), leading to \( x=2 \) or \( x=-\frac{3}{5} \). For \( (2 x-3)(x+2)-5(x+2)=0 \), we can simplify to \( (x+2)(2x-3-5)=0 \), or \( (x+2)(2x-8)=0 \) which gives us solutions \( x=-2 \) or \( x=4 \). The equation \( 5(x-7)^{2}-(x-7)=0 \) can be set as \( (x-7)(5(x-7)-1)=0 \) leading to \( x=7 \) or \( x=7-\frac{1}{5} \), yielding \( x=7 \) or \( x=\frac{34}{5} \). Finally, \( (x-3)^{2}=x(x-3) \) can be expanded and rearranged to \( (x-3)^{2}-x(x-3)=0 \), leading to \( (x-3)(x-3-x)=0 \), giving \( x=3 \) or \( x=0 \). All those solutions showcase the unique beauty of quadratic equations and reveal how they can embody unexpected twists and turns, much like a good plot in a novel! Quadratic equations have a long-standing history in mathematics, tracing back to ancient civilizations like the Babylonians and Greeks who were among the first to study them. The standard form \( ax^2 + bx + c = 0 \) emerged over centuries, culminating in the quadratic formula that we still use today to find solutions efficiently. In the modern world, quadratic equations find real-world applications in various fields such as physics to model projectile motion, finance to calculate profit margins, and engineering to design curves and structures. Understanding how to solve them can open up a world of possibilities in both academic and professional realms!