Today's soundtrack is Lamb of God: Ashes of the Wake, a solid metalcore album that really showcases drummer Chris Adler's chops.
This evening, I'm learning about the quadratic formula, a formula generated by completing the square. We can use the quadratic formula "to solve any quadratic equation" (Pearson, Pre-Calculus 11, p. 213).
The quadratic equation codifies what we do when we complete the square as we solve a quadratic equation. When we are given any quadratic equation, we can use the quadratic formula to solve it thus:
1) Ensure the equation is in the form ax² + bx + c = 0, then compare the quadratic equation with its archetype to determine the values of a, b, and c
2) Substitute the values into the quadratic formula: x = (-b ± √[b² - 4ac] ) / 2a
3) Simplify values
4) Convert the radical to a mixed radical
5) Factor the numerator
6) Simplify by dividing the numerator by the denominator
It's important to note that in this equation, x ∈ ℝ; if we get an answer that isn't a real number, then the "equation has no real roots" (p. 214).