SOLVING QUADRATIC EQUATIONS

Note:


Solve for x in the following equation.

Problem 4.1D tex2html_wrap_inline155 tex2html_wrap_inline352





Answer: tex2html_wrap_inline155 tex2html_wrap_inline354





Solution: tex2html_wrap_inline155 The equation is already equal to zero.





Method 1: tex2html_wrap_inline155 Factoring

The equation cannot be easily factored.





Method 2: tex2html_wrap_inline155 Completing the square

Subtract tex2html_wrap_inline356 from both sides of the equation.

eqnarray60

Add tex2html_wrap_inline358 to both sides of the equation.

eqnarray84

Factor the left side and simplify the right side.

eqnarray96

Take the square root of both sides of the equation,

eqnarray104

Subtract tex2html_wrap_inline360 from both sides of the equation.

eqnarray113

and

eqnarray124





Method 3: tex2html_wrap_inline155 Quadratic Formula

The quadratic formula is tex2html_wrap_inline362

In the equation tex2html_wrap_inline352 , a is the coefficient of the tex2html_wrap_inline366 term, b is the coefficient of the x term, and c is the constant. Simply insert 51 for a, tex2html_wrap_inline372 for b, and tex2html_wrap_inline374 for c in the quadratic formula and simplify.

eqnarray151

and

eqnarray180





Method 4: tex2html_wrap_inline155 Graphing

Graph y= the left side of the equation or

eqnarray188

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

eqnarray188

crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation. The x-intercepts are -0.398858303987 and tex2html_wrap_inline388 The answers are -0.398858303987 and tex2html_wrap_inline392





Check these answers in the original equation.

Check the answer x=-0.398858303987 by substituting -0.398858303987 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 -0.398858303987 for x, then x=-0.398858303987 is a solution.





Check the solution x=-0.156697251568 by substituting -0.156697251568 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 -0.156697251568 for x, then x=-0.156697251568 is a solution.





The solutions to the equation tex2html_wrap_inline352 are and





Comment: tex2html_wrap_inline155 You can use the solutions to factor the original equation.

For example, since tex2html_wrap_inline428 , then tex2html_wrap_inline430 , and tex2html_wrap_inline432

Since tex2html_wrap_inline434 , then tex2html_wrap_inline436 , and tex2html_wrap_inline438

Since the product

eqnarray264

and tex2html_wrap_inline352 , then we can say that

eqnarray279

This means that

eqnarray294

are factors of

eqnarray304


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Author:Nancy Marcus

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