Solving Quadratic Equations in One Variable Using Factoring | Math

January 26, 2015 thetasctest
They're all working towards individual goals A quadratic equation is a specific type of equation, different than the linear equations we’ve worked with before. To be more specific, a quadratic equation is any equation that can be written in the form: ax2 + bx + c = 0, where a, b, and c are coefficients and a ≠ 0. (If a = 0, the equation is a linear equation.) Solving quadratic equations is a high emphasis topic on the TASC Test Assessing Secondary Completion™ Mathematics subtest. Though solving a problem like this may seem complex, specific tools can help you work through any quadratic equation as you prepare for the Math subtest. In fact, there are four options for solving quadratic equations in one variable. Depending on how much information – and what information – is provided within the equation, you can take any of these approaches. Here, we’ve outlined how to factor quadratic equations. We chose to focus on factoring because it’s typically understood to be the easiest way to solve a quadratic equation. It’s also one of the most common methods.

Factoring

To factor, you must follow these steps:
  1. Write your Equation in the form ax2 + bx + c = 0. You can do this by rearranging the formula, as we discussed in our earlier post. You’ll remember this can mean a number of things. You might have to use the distributive property, combine like terms, move terms to different sides of the equation, or apply the order of operations.
  2. Factor your equation.
  3. Let each factor equal 0, and solve.
 

An Example

  Walk through solving one example of a quadratic equation:   (2x + 3)x = 5 First, notice the x is against the parenthesis. This means that it is being multiplied and you have to distribute it. 2x2 + 3x = 5   Then, you want to have all of your numbers on one side of the equation, to make sure it is in the right form. 2x2 + 3x = 5 (2x2 + 3x) - 5 = (5) – 5 2x2 + 3x - 5 = 0   Now that it’s in the right form, you can factor it. (2x + ?)(x - ?) = 0   Here, the two unknown numbers (?) must multiple to equal (-5) when you use the Distributive Property to multiply out. The first two terms must multiply to equal 2x2, and the middle products must equal 3x. Therefore, the answer must be: (2x + 5)(x - 1) = 0   because when factored out we get the original: 2x2 – 2x + 5x – 5 = 0 2x2 + 3x – 5 = 0   Using our complete factors, we can apply the Zero Product Law, which allows us to state: (2x + 5) = 0   and   (x - 1) = 0   To solve the quadratic equation, we can solve these two equations: (2x + 5) – 5 = 0 2x = -5 (2x)/2 = (-5)/2 x = (-5)/2   and   (x - 1) + 1 = 0 x = 1   Therefore, we have found the roots of the quadratic equation and can graph the results. You may only be asked for the roots on the math subtest, or you may be asked to identify what the graph looks like (in which you’ll need to find the roots to identify where the line passes through the plots). It is helpful to note that quadratic equations always make nice curves.   The sources we have included here also describe the other methods for solving quadratic equations, including: Extracting Square Roots, Completing the Square, and Using the Quadratic Formula. Have questions about factoring or any of these additional methods? Leave a comment below and we’ll help you!  
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