How to Find Solutions Roots of an Equation

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:

Factoring

Completing the Square

Quadratic Formula

Graphing
All methods start with setting the equation equal to zero.

Solve for x in the following equation.

Example 1: text2html_wrap_inline253

The equation is already set to zero.

Method 1: text2html_wrap_inline253 Factoring

eqnarray60

eqnarray64

Method 2: text2html_wrap_inline253 Completing the square

Divide both sides of the equation by 2.

Add to both sides of the equation.

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 :

Add to both sides of the equation :

Method 3: text2html_wrap_inline253 Quadratic Formula

The quadratic formula is tex2html_wrap_inline331

In the equation ,a is the coefficient of the term, b is the coefficient of the x term, and c is the constant. Substitute 2 for a, -1 for b, and -1 for c in the quadratic formula and simplify.

eqnarray189

eqnarray196

Method 4: text2html_wrap_inline253 Graphing

Graph y= the left side of the equation or 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 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 1 and one at .

The answers are 1 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 x=1 by substituting 1 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 for x, then x=1 is a solution.

Check the solution tex2html_wrap_inline373 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 tex2html_wrap_inline373

is a solution.

The solutions to the equation are 1 and

If you would like to work another example, click on Example.

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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Author:Nancy Marcus

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