SOLVING QUADRATIC EQUATIONS

Note:




Solve for x in the following equation.

Problem 4.2b:text2html_wrap_inline253 tex2html_wrap_inline326


Answer:text2html_wrap_inline253 tex2html_wrap_inline328


Solution:

Remove the denominators in the original equation by multiplying both sides by 16.

eqnarray30

eqnarray38

eqnarray46



Set the equation tex2html_wrap_inline330 equal to zero by subtracting 66x and adding 509 to both sides of the equation.

eqnarray53

eqnarray57

eqnarray61





Method 1:text2html_wrap_inline253 Factoring

The equation tex2html_wrap_inline334 can be written as

eqnarray68



The only way a product can equal zero is for at least one of the factors to have a value of zero:

eqnarray72





Method 2:text2html_wrap_inline253Completing the square

Add 3 to both sides of the equation tex2html_wrap_inline336 .

eqnarray84



Divide both sides by 16 :

eqnarray89



Add tex2html_wrap_inline338 to both sides of the equation:

eqnarray105



Factor the left side and simplify the right side:

eqnarray117



Take the square root of both sides of the equation :

eqnarray125



Add tex2html_wrap_inline340 to both sides of the equation :

eqnarray134

and

eqnarray148





Method 3:text2html_wrap_inline253Quadratic Formula

The quadratic formula is tex2html_wrap_inline342

In the equation tex2html_wrap_inline344 , a is the coefficient of the tex2html_wrap_inline346 term, b is the coefficient of the x term, and c is the constant. Simple insert 16 for a, -2 for b, and -3 for c in the quadratic formula and simplify

.

eqnarray173

eqnarray179

and

eqnarray187





Method 4:text2html_wrap_inline253Graphing

Graph tex2html_wrap_inline350 (formed by subtracting the right side of the original equation from the left side of the original equation. Graph y=0 (the x-axis).

What you will be looking for is where the graph of tex2html_wrap_inline354 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 located at tex2html_wrap_inline358 and tex2html_wrap_inline360 This means that there are two real answers: tex2html_wrap_inline364 and tex2html_wrap_inline366 The answers are tex2html_wrap_inline368 and tex2html_wrap_inline360 These answers may or may not be solutions to the original equation. You must check the answers with the original equation.




Check these answers in the original equation.

Check the solution tex2html_wrap_inline364 by substituting tex2html_wrap_inline358 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_inline358 for x, then tex2html_wrap_inline364 is a solution.



Check the solution tex2html_wrap_inline392 by substituting tex2html_wrap_inline394 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_inline394 for x, then tex2html_wrap_inline392 is a solution.

The solutions to the equation tex2html_wrap_inline404 are tex2html_wrap_inline394 and tex2html_wrap_inline408



If you would like to review the solution to 4.2c, click on Solution

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