SOLVING QUADRATIC EQUATIONS

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
3. Quadratic Formula
4. Graphing
All methods start with setting the equation equal to zero.

Solve for x in the following equation.

Example 5: tex2html_wrap_inline176

The equation is already equal to zero.

Method 1: Factoring

We will not use this method because the left side of the equation is not easily factored .

Method 2: Completing the square

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.

and

Method 3: Quadratic Formula

The quadratic formula is tex2html_wrap_inline90 .

In the equation tex2html_wrap_inline92 , a is the coefficient of the term, is the coefficient of the x term, and c is the constant. Simply insert 1 for a, for b, and for c in the quadratic formula and simplify

eqnarray31

eqnarray48

eqnarray60

and

Method 4: Graphing

Graph y= the left side of the equation or tex2html_wrap_inline349 and graph y= the right side of the equation or y=0. The graph of y=0 is nothing more than the x-axis. So what you will be looking for is where the graph of tex2html_wrap_inline357 crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation. There are two x-intercepts located at 0.638491982474 and -1.30515864914. This indicates that there are two real answers.

Check these answers in the original equation.

Check the answer x=0.638491982474 by substituting 0.638491982474 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 0.638491982474 for x, then x=0.638491982474 is a solution.

Check the answer x=-1.30515864914 by substituting -1.30515864914 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 -1.30515864914 for x, then x = -1.30515864914 is a solution.

Comment: You can use the solutions to factor the original equation.

For example, since tex2html_wrap_inline395 , then tex2html_wrap_inline397 and tex2html_wrap_inline399

Since tex2html_wrap_inline401 then tex2html_wrap_inline403 and tex2html_wrap_inline405

Since the product tex2html_wrap_inline407 and tex2html_wrap_inline409 , then we can say that tex2html_wrap_inline411

This means that tex2html_wrap_inline413 and tex2html_wrap_inline415 are factors of tex2html_wrap_inline417

If you would like to test yourself by working some problems similar to this example, click on Problem.
If you would like to go back to the equation table of contents, click on Contents.

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