SOLVING QUADRATIC EQUATIONS



Note:




Solve for x in the following equation.

Example 1: tex2html_wrap_inline155 tex2html_wrap_inline275

The equation is already set to zero.

If you have forgotten how to manipulate fractions, click on Fractions 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 2.

eqnarray41







Method 1: tex2html_wrap_inline155 Factoring

The equation tex2html_wrap_inline277 is not easily factored. Therefore, we will not use this method.







Method 2: tex2html_wrap_inline155 Completing the square

Add 10 to both sides of the equation

eqnarray56


Add tex2html_wrap_inline279 to both sides of the equation:

eqnarray65


Factor the left side and simplify the right side:

eqnarray69


Take the square root of both sides of the equation :

eqnarray74


Add 16 to both sides of the equation :

eqnarray78







Method 3: tex2html_wrap_inline155 Quadratic Formula

The quadratic formula is tex2html_wrap_inline281

In the equation tex2html_wrap_inline283 ,a is the coefficient of the tex2html_wrap_inline285 term, b is the coefficient of the x term, and c is the constant. Substitute 1 for a, -32 for b, and -10 for c in the quadratic formula and simplify.

eqnarray98


eqnarray107







Method 4: tex2html_wrap_inline155 Graphing

Graph the left side of the equation, tex2html_wrap_inline289 and graph the right side of the equation, tex2html_wrap_inline291 The graph of tex2html_wrap_inline293 is nothing more than the x-axis. So what you will be looking for is where the graph of tex2html_wrap_inline295 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 32.309506 and one at -0.309506.

The answers are 32.309506 and tex2html_wrap_inline307 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=32.309506 by substituting 32.309506 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 32.309506 for x, then x=32.309506 is a solution.

Check the solution x=-0.309506 by substituting -0.309506 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 -0.309506 for x, then x= - 0.309506 is a solution.





The solutions to the equation tex2html_wrap_inline337 are 32.309506 and - 0.309506.





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


Since tex2html_wrap_inline343


Since tex2html_wrap_inline345


The product tex2html_wrap_inline347


Since tex2html_wrap_inline349


then we could say


displaymath263


However the product of the first terms of the factors does not equal tex2html_wrap_inline351


Multiply tex2html_wrap_inline353


Let's check to see if tex2html_wrap_inline357


displaymath264


displaymath265


displaymath266


displaymath267


displaymath268


The factors of tex2html_wrap_inline359 are tex2html_wrap_inline361 , and tex2html_wrap_inline363






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


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Author: Nancy Marcus

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