actor completely. \[ 3 x^{2}-20 x+25 \]

To factor the quadratic expression \( 3x^{2} - 20x + 25 \), we look for two numbers that multiply to give \( a \cdot c = 3 \cdot 25 = 75 \) and add up to \( b = -20 \). In this case, the pairs that work are -15 and -5. Rewriting the expression, we have: \[ 3x^{2} - 15x - 5x + 25 \] Next, we group and factor by parts: \[ (3x^{2} - 15x) + (-5x + 25) = 3x(x - 5) - 5(x - 5) \] This gives us: \[ (3x - 5)(x - 5) \] The complete factorization is: \[ (3x - 5)(x - 5) \] Now we have it all ready for any further explorations! In the world of quadratic expressions, factoring not only simplifies solving equations but can also unveil different graph characteristics! Each factor can indicate the x-intercepts of a corresponding parabola. So, once you've factored it, you can easily sketch the graph for further insights, all while uncovering the beauty of mathematics. Have you ever felt lost in quadratic equations? No worries! A common mistake is failing to recognize when to apply the quadratic formula. Before resorting to that, always check if the expression can be factored easily. It saves you time and helps you develop a keener intuition for numbers! Start by looking for a common factor or testing if the expression can be easily grouped first. Happy factoring!