Quadratic Equation

A quadratic equation is an equation of the form:, where a, b, and c are real numbers with a0. (The degree of the equation is 2) (The highest exponent of a variable is 2)

Solving Equations of the form: 
 

Solve the equation:	Given Problem.
	Add one to both sides of the equation.
	Divide both sides of the equation by 4.
	Take the square root of both sides of the equation.
	Solution set. Check both roots.

Here we need to introduce two new definitions:

Definitions:

Zero Product Property	When ever you have a product which is set equal to zero, then each and every factor can cause the product to be zero.  (or any combination thereof)
Extraneous Roots	Roots (or solutions) to a quadratic equation which have been obtained by correct algebraic steps, but which do not check in the original equation. Extraneous roots are not solutions.

Solving Quadratic Equations by Factoring:
 

Solve the equation:	Given Problem.
	Rewrite the equation so that it is set equal to zero. Do this by adding 24 to both sides of the equation.
	Factor the left side of the equation.
	Because of the Zero Product Property, each factor can be set equal to zero. These equations can then be solved.
	It is important to check solutions to quadratic equations. Solution techniques can sometimes result in extraneous roots. In this case both answers check.
	The solution set.

Definition:
 

Solving Quadratic Equations by Completing the Square:
 

Solve the equation:	Given Problem. Please note that this equation is not factorable. Because we cannot factor it, we must use a new technique, namely "completing the square."
	We begin this technique by "removing" the constant from the left side. In this case that means adding 25 to both sides of the equation.
	This is the key step. We want to complete the incomplete trinomial on the left so that it becomes a perfect square trinomial. First, we must make sure that the coefficient of the x squared term is 1. If it is not, then we would have to divide both sides of the equation by the coefficient as our next step. Once we are sure of this coefficient being one, we take half of the coefficient and then square that. To follow ERAA, we add this quantity to both sides of the equation.
	We express the left side as a binomial squared, and simplify the right side.
	We now take the square root of both sides of the equation. Remember to take both square roots on the right side.
	Now we subtract 5 from both sides of the equation.
	After using a calculator to check both roots, we have our solution set.

Completing the Square Flash version

Using this technique of completing the square, we will take the general quadratic equation, and develop a formula for the solutions:
 

	This the the general quadratic equation that we defined earlier in this lesson.
	Just as in the last example, we "vacate" the constant "c" from the left side. We do this by subtracting "c" from both sides of the equation.
	Here we do not have a "1" for our coefficient of our x square term, so we must begin by dividing both sides of the equation by a.
	Now we complete the square by taking half of the x coefficient term and squaring it. To follow ERAA, we add this to both sides of the equation.
	This step shows how the right side of the equation can be simplified. We find a common denominator, and combine the two fractions.
	We now express the left side of the equation as a binomial squared.
	Next, we take the square root of both sides of the equation.
	Next, we subtract  from both sides of the equation.
	Finally, we simplify the right side, taking advantage of the common denominator.

Definition:

Quadratic Formula

Given a quadratic equation in the form: , and that  ,then the solutions of the equation will always be: .

Solving Quadratic Equations Using the Quadratic Formula:
 

Solve the equation:	Given problem. Notice that the quadratic equation is not factorable. When this is the situation you have two choices. You could use completing the square, or the quadratic formula. This is not surprising because the quadratic formula came from using completing the square.
	Make sure the equation is in standard form (set equal to zero). Identify the coefficients a, b, and c.
	Substitute a, b, and c into the quadratic formula.
	Simplify under the radical.
	Simplify the root.
	Factor the numerator and cancel the twos.
	The solution set. (check using a calculator)

Some Notes form Class