Algebra: Solving Quadratic Equations by Factoring

Previously we have learned how to solve linear equations, ax + b = 0, now we will outline a technique that can be used to solve factorable quadratic equations of the form ax^2 + bx + c = 0. Unlike linear equations, which usually have one solution, quadratic equations can have up to two solutions. We begin with the zero product property:

This property is the key to solving quadratic equations by factoring. Basically, if the quadratic equation is equal to zero you factor it, then set each factor equal to zero, and solve.

Tip: You can always check your answers by substituting them into the original equation to see if a true statement results. 

  

Solve.

Note: When one solution is obtained, as in the previous problem, we say that the solution is a double root. This technique requires the zero product property, so you first make sure that the quadratic equation is set equal to zero before factoring. Do this by adding all terms to one side of the equation. The check is usually optional.

A common mistake is to set each factor to 56 here - which leads to incorrect results. You must have zero on one side of the equation for this technique to work.

Solve.

Clearing fractions from our equations can be done by multiplying both sides of our equation by the LCM of the denominators. After doing this, the equation becomes a bit easier to solve.

Solve.

Finding equations given the solutions requires you to work the entire process in reverse. Given the solutions you can find factors that can be multiplied to obtain a quadratic equation.

Find a quadratic equation with the given solution set.

You can find much more in the free to read online open Elementary Algebra textbook available on the Flat World Knowledge website:

[ Free Algebra Textbook ]

UPDATE: Here are some corresponding exercises. Enjoy.

[ Exercises PDF ] [ Download Source DOCX ]

---