Quadratic Equations

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Solving Quadratic Equations by Factoring

Recall the standard form of a quadratic equation is ax2 + bx + c = 0, where a  is not zero.  One way to solve a quadratic equation is by factoring.  To solve by factoring, you must use the Zero Product Property.

Zero Product Property

For any real numbers a and b, if ab = 0,
then either a = 0, or b = 0, or both.

An algebraic method for solving quadratic equations may be the preferred approach.  We can often solve quadratic equations by using factoring and the Zero Product Property.

example #1:




The zeros of the function are {0, 5}.
In other words (0,0) and (5,0) are the points on the graph where the parabola crosses the x-axis.

 

example #2:




The zeros of the function are {–6, 3}.
In this example, the parabola has two solutions located at (–6, 0) and (3, 0).

 

example #3:



The zeros of the function are located at {5/3 -7/2}.  Her is an example where a graph would not reveal exact solutions in an efficient manner.  The zeros are rational numbers instead of integers.

You can use the following steps to solve equations using the zero product property.

  1. Get zero on one side of the equation using the addition property.
  2. Factor the expression on the other side of the equation.
  3. Set each factor equal to zero.
  4. Solve each equation.

Solving Quadratics By Finding Square Roots

 

example #4:
Rewrite in the form ax2 = c.
Isolate x2.
Simplify.
Take the square root of each side.
Simplify radical expression completely.

 


The zeros of the function are :

Solving By Graphing

Solutions for Quadratic Equations

Not every quadratic equation can be solved by factoring or by taking the square root.  It is possible to solve some quadratic equations in standard form by graphing the related quadratic function y = ax2 + bx + c.  When the graph of the function intersects the x-axis, the value of the function is zero.  This point is commonly referred to as an x-intercept.  We will now call these x-values zeros of the function.  A zero of a a function is a solution of the quadratic equation .

The last area to explore in this section are the three cases or possible outcomes that can occur when solving quadratic equations.

There are three possible outcomes when solving a quadratic equation.

  1. Two real solutions,

  2. One real solution, or

  3. No real solutions.

Graphically, these outcomes look like the following:

Two real solutions
Notice this parabola  crosses the x-axis twice.  The x-coordinates of these intersection points are called the zeros of the function.
In this example, the zeros are located where x = – 3 and where x = 1.

 

 
One real solution
When a quadratic equation has one real solution, it really has two solutions that are the same number.  In this example the zero is where x = 1.

 

 
No real solutions

 

When a quadratic function does not cross the x-axis it is because no real number solutions exist.