Advantages and disadvantages of using journals in mathematics?

Answer 1

Somebody (possibly in seventh-century India) was solving a lot of quadratic equations by completing the square. At some point, he noticed that he was always doing the exact same steps in the exact same order for every equation. Taking advantage of the one of the great powers and benefits of algebra (namely, the ability to deal with abstractions, rather than having to muck about with the numbers every single time), he made a formula out of what he'd been doing:

The Quadratic Formula: For ax2 + bx + c = 0, the value of x is given by

The nice thing about the Quadratic Formula is that the Quadratic Formula always works. There are some quadratics (most of them, actually) that you can't solve by factoring. But the Quadratic Formula will always spit out an answer, whether the quadratic was factorable or not.

I have a lesson on the Quadratic Formula, which gives examples and shows the connection between the discriminant (the stuff inside the square root), the number and type of solutions of the quadratic equation, and the graph of the related parabola. So I'll just do one example here. If you need further instruction, study the lesson at the above hyperlink.

Let's try that last problem from the previous section again, but this time we'll use the Quadratic Formula:

• Use the Quadratic Formula to solve x2 - 4x - 8 = 0.

Looking at the coefficients, I see that a = 1, b = -4, and c = -8. I'll plug them into the Formula, and simplify. I should get the same answer as before: