Problem 68 - Introductory Algebra for College Students

Question

If you are given a quadratic equation, how do you determine which method to use to solve it?

Step-by-Step Solution

Answer

The method for solving a quadratic equation can be chosen based on the equation given. Use factoring when the equation can be easily factored, use the quadratic formula when the equation is complex and cannot be factored, and use completing the square when the equation is in the form \(a(x-h)^2 = k\).

Step 1: Understanding the Quadratic Equation

A quadratic equation is an equation which can be written in the form \(ax^2 + bx + c = 0\), where a, b, and c are constants and \(a \neq 0\). The solutions or roots of this equation can be real or complex numbers.

Step 2: Exploring Different Methods

There are three commonly used methods to solve quadratic equations: Factoring, Quadratic formula, and completing the square.

Step 3: Choosing Factoring

Factoring is the best method to use when the quadratic equation can be easily factored. It is the fastest and easiest method. For example, if we have the quadratic equation \(x^2 - 3x - 4 = 0\), it can be factored into \((x-4)(x+1) = 0\). Therefore, factoring is the best method to use here.

Step 4: Choosing Quadratic Formula

The quadratic formula is used when the quadratic equation cannot be easily factored. The quadratic formula is \(-b ± \sqrt{b^2 - 4ac} \div 2a\). It can solve any quadratic equation.

Step 5: Choosing Completing the Square

Completing the square is typically used when the equation is in the form \(a(x-h)^2 = k\), because in this form, it is easy to see the solutions. However, this method can be more complex and requires more steps than the other two methods.