Solutions to Quadratic Equations
There are several ways we can talk about solutions to quadratic equations. Solutions are also called roots or zeros, because they represent x-values to the equation that make y equal to zero. Below is a graphical representation to solutions to a quadratic equation:
Quadratic equations with have either one, two, or zero real solutions. We can relate this to the graphical interpretation of solutions. A parabola will intersect the x-axis no more than two times. If a parabola never intersects the x-axis, it has no real solutions.
Zero Factor Property of Multiplication
If any factor of an algebraic expression equals zero, the entire expression has a value of zero. This is because any quantity multiplied by zero is zero. This fact comes in handy when solving quadratic equations written in standard form. We can simply set each factor equal to zero, and solve simpler equations to find solutions to the quadratic. Consider the example below:
By setting each factor equal to zero and solving for x, we have found two solutions to the quadratic. When x = –3, the quadratic will equal zero, because (x + 3) = 0 at x = –3. Similarly, when x = 7, the quadratic will equal zero, because (x – 7) = 0 when x = 7.
Use the Zero Factor Property of Multiplication to solve quadratic equations by setting each factor equal to zero. Solving for x when each factor is set equal to zero will reveal solutions to the quadratic.
Solving a Quadratic with No Constant Term
When a quadratic is set equal to zero, but has no constant term, making use of the Zero Factor Property is still helpful, but there is some work that needs to be done first. Without the constant term, we know that all of the terms in the quadratic share a factor of x. This means we can factor it out, and then apply the Zero Factor Property.
Solving a Quadratic with No x-Term
If a quadratic equation is missing an x-term, using the Zero Factor Property is not as helpful. Instead, we can solve using standard algebraic techniques for solving any equation. The important thing to remember here is that when we apply the square root to a quantity squared, we must include both positive and negative values of the root because a negative value squared is also positive.
Be sure to include plus or minus when taking the square root in equations. In our example above, both and evaluate to 7 when squared.