Note:
Solve for x in the following equation.
Example 3:
The equation is already set to zero.
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 how to manipulate fractions, click on Fractions for a review.
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 56. All the denominators 8,
28, and 7 divide into 56 evenly.
Method 1:Factoring
The equation can be factored as follows:
Method 2:Completing the square
Divide both sides of the equation by 21.
Add to both sides of the equation:
Simplify:
Add to both sides of the equation:
Factor the left side and simplify the right side:
Take the square root of both sides of the equation:
Subtract from both sides of the equation:
Method 3:Quadratic Formula
The quadratic formula is
In the equation ,a is the coefficient of the term, b is the coefficient of the x term, and c is the constant. Substitute for a, for b, and for c in the quadratic formula and simplify.
Method 4:Graphing
Graph the equation, (formed by subtracting the right side of the equation from the
left side of the equation). Graph (the x-axis). What you
will be looking for is where the graph of 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 and one at .
The answers are and 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 by substituting 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 for x,
then is a solution.
Check the solution by substituting 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.
The solutions to the equation are and
Comment:Recall that when we solved this equation by factoring, we factored
the expression not the original expression
The product of the
factors of 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.
Product of factors
equals What number do I multiply 21 by to get
Let's us see if equals the original
The factors of are
We have illustrated that the solutions to the equation are
and
If you want to verify that you know how to work this type of problem by
testing yourself over problems similar to the one above, click on Problem
If you would like to go back to the equation table of contents, click
on Contents
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