Solving Quadratic Equations
A quadratic equation is a polynomial of degree 2. It always has an x2 term, but no larger exponents. Another way to write that, with more letters but fewer words, is that a quadratic equation has the form y = ax2 + bx + c, where a ≠ 0.
Solving an equation means to find all values of x that, when plugged into the equation, will make the whole thing equal zero. That is, what values will make ax2 + bx + c = 0? It's like a puzzle, and you're looking for a five-letter word that means 'the solution to the equation.' Those values of x are the roots of the equation.
Sometimes, solving a quadratic equation is easy. Take x2 = 4, for instance. You can take the square root of both sides of the equation, finding x = 2 and x = -2 as the solutions. Other times, you can factor the quadratic equation into a squared term: x2 + 2x + 1 = 0 becomes (x + 1)2 = 0, giving a solution of x = -1.
But quadratic equations don't always play nice. What can we do when they start getting unruly? We'll look at two techniques that can be used on any quadratic equation. We'll also go over a method for predicting the number and type of solutions you'll find for any particular equation. Buckle up.