Solving Quadratics Using the Quadratic Formula

If you say “quadratic formula”, many people have probably heard of it but do not know what it is used for or what the actual formula is. This lesson will use the formula and discuss how the formula is used to solve problems.

The purpose of the quadratic formula is to solve a quadratic equation. It is important to note the distinction between these two. The quadratic formula is a tool. A quadratic equation is usually given to you specifically for the purpose of solving it. When solving a quadratic equation, you are finding the roots of that equation. The roots of the equation will tell you where the graph touches or crosses the x – axis. If you need to know more about quadratic equations or other terminology, click here for an introductory lesson.

Many people may be able to memorize the quadratic formula as ^. However, you have to know what all those letters mean in order to use the formula properly. The quadratic formula can only be applied to a quadratic equation that is in the standard form of . If an equation is not in this form, it must be manipulated until it is in standard form - only then can the quadratic formula be applied.

Example Group #1

First we will use the quadratic formula to determine the solutions for x in equations that could also be solved by using factoring methods.

	#1: Solve First see if the equation is in Standard form. It is, so we proceed. Identify a, b, and c in the equation. a = 1, b = -4, and c = -5 Substitute the values of a, b, and c into the quadratic formula. Be careful with parentheses and negative numbers. Simplify this expression to determine the values for x.

	What is your answer?

	#2: Solve First see if the equation is in standard form. It is not, so we move the –8 to the left side of the equation. Identify a, b, and c in the equation. a = 1, b = -6 and c = 8 Substitute the values of a, b, and c into the quadratic formula. Simplify this expression to determine the values for x.

	What is your answer?

Example Group #2

In each of the first two examples, the solutions turn out to be integer values. When this is the case, it means that you could have used factoring as a method for solving the quadratic equation. The quadratic formula is most useful for equations that do not factor. In situations where you cannot factor the quadratic equation, the only way to find the roots is to use the quadratic formula.

	3: Solve First see if the equation is in standard from. It is, so we proceed. Identify a, b, and c in the equation. a = 2, b = 4, and c = -5 Substitute the values of a, b, and c into the quadratic formula Simplify this expression to determine the values for x.

	What is your answer?

	4: Solve First see if the equation is in standard from. It is not, so we move the –12x and the –9 to the left side of the equation. Identify a, b, and c in the equation. a = 4, b = 12, and c = 9 Substitute the values of a, b, and c into the quadratic formula and simplify to solve for x.

	What is your answer?

	5: Solve First see if the equation is in standard from. It is, so we proceed. Identify a, b, and c in the equation, substitute their values into the quadratic formula and simplify the expression to solve for x.

	What is your answer?

Whether the solution to the equation is an integer, real number, imaginary or a double root depends on the value under the square root in the quadratic formula

The value of

is called the discriminant and it determines what type of answer you will get.

When is a perfect square, you will get rational answers for your solution.
When the value is not a perfect square, you will get two real number solutions that should be expressed as an exact value rather than a decimal approximation.
As we said earlier, if = 0, then you will only get one rational solution. That value will be a double root.
If the value of is negative, then you will have two imaginary solutions. In other words, your graph will not touch the x–axis.

Examples

	Use the value of the discriminant to determine what type of roots the equation should have.

	What is your answer? one rational root solution two real number solutions two rational root solutions two imaginary root solutions

	Use the value of the discriminant to determine what type of roots the equation should have.

	What is your answer? one rational root solution two real number solutions two rational root solutions two imaginary root solutions