Algebra 1 Factor completely. \( 2 q^{2}-q-10 \) \( (-)(+) \) subctor quas:

To factor the quadratic expression \(2q^2 - q - 10\), we need to look for two numbers that multiply to the product of the coefficient of \(q^2\) (which is 2) and the constant term (which is -10). That's \(2 \times -10 = -20\). We need these two numbers to also add up to the coefficient of \(q\), which is -1. The numbers -5 and 4 fit this requirement since \(-5 + 4 = -1\) and \(-5 \times 4 = -20\). Now, we can rewrite the expression: \(2q^2 - 5q + 4q - 10\) Next, factor by grouping: \(q(2q - 5) + 2(2q - 5)\) Now, we can factor out the common term \((2q - 5)\): \((2q - 5)(q + 2)\) So, the completely factored form is \((2q - 5)(q + 2)\). Now you can double-check your factorization by distributing back to ensure it matches the original expression! Another fun fact: Quadratic expressions like this are not just for math classes! They're used in real-world applications such as physics when calculating trajectories, economics, and even in computer graphics for modeling curves. You'll find that mastering these concepts can open doors to understanding complex scenarios in the real world. Keep practicing, and you'll be a factoring pro in no time!