Note:
Solve for x in the following equation.
Example 4:
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?
and If we choose the set of factors
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.
Method 1:Factoring
The left side of the equation can be factored as follows :
The answers are
Method 2:Completing the square
Divide both sides of the equation by 77 .
Simplify:
Add to both sides of the equation
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:
Add to both sides of the equation:
The answers are
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.
The answers are
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.
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.
Original Equation :
Product of factors
equals What number do I multiply 77 by to
get
Let's us see if equals the original 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
This confirms the factors are correct using equivalence
Algebraically
Since the original expression is , we have shown the factors are correct algebraically.
We have illustrated that the solutions to the equation are
and
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