SOLVING QUADRATIC EQUATIONS

Note:




Solve for x in the following equation.


Example 1: text2html_wrap_inline253 tex2html_wrap_inline321

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 tex2html_wrap_inline323 by 2.

eqnarray80




Add tex2html_wrap_inline325 to both sides of the equation.

eqnarray97




Add tex2html_wrap_inline327 to both sides of the equation:

eqnarray121




Factor the left side and simplify the right side :

eqnarray133




Take the square root of both sides of the equation :

eqnarray141




Add tex2html_wrap_inline329 to both sides of the equation :

eqnarray150







Method 3:text2html_wrap_inline253 Quadratic Formula

The quadratic formula is tex2html_wrap_inline331

In the equation tex2html_wrap_inline333 ,a is the coefficient of the tex2html_wrap_inline335 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 tex2html_wrap_inline341 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_inline341 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 tex2html_wrap_inline353 .

The answers are 1 and tex2html_wrap_inline357 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.

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 tex2html_wrap_inline353 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.

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value tex2html_wrap_inline353 for x, then tex2html_wrap_inline373 is a solution.




The solutions to the equation tex2html_wrap_inline385 are 1 and tex2html_wrap_inline389






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Author:Nancy Marcus

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