SOLVING QUADRATIC EQUATIONS



Note:




Solve for x in the following equation.

Example 4:tex2html_wrap_inline155tex2html_wrap_inline620


The equation is already set to zero.


If you have forgotten how to manipulate fractions, click on Fractions for a review.


Almost all precalculus students hate fractions before they renew their love for them. The following steps will transform the equation into an equivalent equation without fractions.


If you have forgotten what equivalent means, think of a dollar. You can represent the dollar with a dollar bill, 10 dimes, 20 nickels, or 100 pennies. All of these are equivalent because all have the value of a dollar. Got it. If not, click on Equivalence 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 462. All the denominators 6, 231, and 77 divide into 462 evenly.


How do we know this? tex2html_wrap_inline632 tex2html_wrap_inline634 and tex2html_wrap_inline636 If we choose the set of factors

eqnarray59


the set of factors of each denominator can be found in this set. This means that each of the denominators divides evenly into the product 462.



eqnarray65


eqnarray78


eqnarray89







Method 1:tex2html_wrap_inline155Factoring

The left side of the equation tex2html_wrap_inline640 can be factored as follows :

eqnarray103


eqnarray109


The answers are tex2html_wrap_inline642







Method 2:tex2html_wrap_inline155Completing the square

Divide both sides of the equation tex2html_wrap_inline644 by 77 .


eqnarray135



Simplify: tex2html_wrap_inline650



Add tex2html_wrap_inline652 to both sides of the equation tex2html_wrap_inline654



Add tex2html_wrap_inline656 to both sides of the equation:


eqnarray186



Factor the left side and simplify the right side:


eqnarray205



Take the square root of both sides of the equation:


eqnarray222



Add tex2html_wrap_inline658 to both sides of the equation:


eqnarray240


eqnarray247



The answers are tex2html_wrap_inline642







Method 3:tex2html_wrap_inline155Quadratic Formula

The quadratic formula is tex2html_wrap_inline662

In the equation tex2html_wrap_inline664 ,a is the coefficient of the tex2html_wrap_inline666 term, b is the coefficient of the x term, and c is the

constant. Substitute tex2html_wrap_inline670 for a, tex2html_wrap_inline672 for b, and tex2html_wrap_inline674 for c in the quadratic formula and simplify.



eqnarray292


eqnarray300


eqnarray309


eqnarray318



The answers are tex2html_wrap_inline642







Method 4:tex2html_wrap_inline155Graphing


Graph the equation, tex2html_wrap_inline678 (formed by subtracting the right side of the equation from the left side of the equation). Graph tex2html_wrap_inline680 (the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline682 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 tex2html_wrap_inline686 and one at tex2html_wrap_inline688 .


The answers are tex2html_wrap_inline692 and tex2html_wrap_inline694 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 tex2html_wrap_inline700 by substituting tex2html_wrap_inline692 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_inline692 for x, then tex2html_wrap_inline700 is a solution.

Check the solution tex2html_wrap_inline712 by substituting tex2html_wrap_inline714 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_inline714 for x, then tex2html_wrap_inline712 is a solution.







The solutions to the equation tex2html_wrap_inline155 tex2html_wrap_inline724 tex2html_wrap_inline155 are tex2html_wrap_inline155 tex2html_wrap_inline692 and tex2html_wrap_inline155 tex2html_wrap_inline728







Comment:tex2html_wrap_inline155 Recall that when we solved this equation by factoring, we factored the expression tex2html_wrap_inline730 not the original expression tex2html_wrap_inline732 The product of the factors of tex2html_wrap_inline734 does not equal the original expression because the product of the first terms of the factors must equal the first term of the original expression.

We need to add a constant factor.


Original Equation : tex2html_wrap_inline736


Product of factors tex2html_wrap_inline738 equals tex2html_wrap_inline740 What number do I multiply 77 by to get tex2html_wrap_inline744

eqnarray480

Let's us see if tex2html_wrap_inline746 equals the original tex2html_wrap_inline748 We can do this using equivalence (evaluating both expressions by the same number). If the answers are the same, you have factored the original expression correctly. We can also this by algebraically by multiply the factors out and seeing if they equal the original expression


Equivalence:

Evaluate both expressions at x=2


eqnarray509


eqnarray517


eqnarray520


eqnarray517


This confirms the factors are correct using equivalence


Algebraically


eqnarray554



Since the original expression is tex2html_wrap_inline736 , we have shown the factors are correct algebraically.



We have illustrated that the solutions to the equation tex2html_wrap_inline754 are tex2html_wrap_inline756 and tex2html_wrap_inline758








By now you should be an expert on this type of problem. If you want a verification of this fact, test yourself over problems similar to the one above by clicking 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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