Problem 161 - Algebra and Trigonometry

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 to solve a quadratic equation depends on the equation itself. It can be factored, solved using a quadratic formula, or completed the square. Factoring is usually the simplest and quickest method, but if factoring isn't possible, then use the quadratic formula. In some specific cases where \( a = 1 \) or the equation is a perfect square, completing the square is recommended.

Step 1: Identify Coefficients

First, identify the coefficients \( a, b, \) and \( c \) from the quadratic equation \( ax^2+bx + c=0 \).

Step 2: Check for Factorability

Check to see if the quadratic equation can be factored. For example, for the equation \( x^2-5x+6=0 \), you can factor it into \( (x-2)(x-3) \). So the roots are \( x=2 \) and \( x=3 \). If you can't factor the equation, proceed to the next step.

Step 3: Use Quadratic Formula

If the equation cannot be factored, you can use the quadratic formula which is \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) to find the roots.

Step 4: Use Completing the Square

Another method is completing the square. This method is best applied when \( a = 1 \) or the quadratic equation is a perfect square.