Note:

• A quadratic equation is a polynomial equation of degree 2.

• The ''U'' shaped graph of a quadratic is called a parabola.

• A quadratic equation has two solutions. Either two distinct real solutions, one double real solution or two imaginary solutions.

• There are several methods you can use to solve a quadratic equation:

1. Factoring
2. Completing the Square
4. Graphing

Solve for x in the following equation.

Example 3:

The equation is already set to zero.

Almost all precalculus students hate fractions before they renew their love for them. The following steps will transform the equation into an equivalent equation without fractions.

If you have forgotten how to manipulate fractions, click on Fractions for a review.

If you have forgotten what equivalent means, think of a dollar. You can represent the dollar with a dollar bill, 10 dimes, 20 nickels, or 100 pennies. All of these are equivalent because all have the value of a dollar. Got it. If not, click on Equivalence for a review.

Remove all the fractions by writing the equation in an equivalent form without fractional coefficients. In this problem, you can do it by multiplying both sides of the equation by 56. All the denominators 8, 28, and 7 divide into 56 evenly.

Method 1:Factoring

The equation can be factored as follows:

Method 2:Completing the square

Divide both sides of the equation by 21.

Add to both sides of the equation:

Simplify:

Add to both sides of the equation:

Factor the left side and simplify the right side:

Take the square root of both sides of the equation:

Subtract from both sides of the equation:

In the equation ,a is the coefficient of the term, b is the coefficient of the x term, and c is the constant. Substitute for a, for b, and for c in the quadratic formula and simplify.

Method 4:Graphing

Graph the equation, (formed by subtracting the right side of the equation from the left side of the equation). Graph (the x-axis). What you will be looking for is where the graph of crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation.

You can see from the graph that there are two x-intercepts, one at and one at .

The answers are and These answers may or may not be solutions to the original equations. You must verify that these answers are solutions.

Check these answers in the original equation.

Check the solution by substituting in the original equation for x. If the left side of the equation

equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

• Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value for x, then is a solution.

Check the solution by substituting in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

• Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value for x, then is a solution.

The solutions to the equation are and

Comment:Recall that when we solved this equation by factoring, we factored the expression not the original expression The product of the factors of does not equal the original expression because the product of the first terms of the factors must equal the first term of the original expression.

We need to add a constant factor.

Product of factors equals What number do I multiply 21 by to get

Let's us see if equals the original

The factors of are

We have illustrated that the solutions to the equation are and

Are you secure in your ability to solve this type of equation? If not and if you would like to work another example, click on Example

If you want to verify that you know how to work this type of problem by testing yourself over problems similar to the one above, click on Problem

[Algebra] [Trigonometry]
[Geometry] [Differential Equations]
[Calculus] [Complex Variables] [Matrix Algebra]

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Author: Nancy Marcus