Note:
Solve for x in the following equation.
Problem 4.4a:
Answer: are the exact answers are approximate answers.
Solution:
Set the equation equal to zero by subtracting 2 from both sides.
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 3.
Method 1:Factoring
The equation is not easily factored. Therefore, we will not use this method.
Method 2:Completing the square
Add 36 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 6 to both sides of the equation:
are the exact answers are approximate answers.
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.
are the exact answers are approximate answers.
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 14.485281 and one at -2.485281.
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 x=14.485281 by substituting 14.485281 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 14.485281 for x, then
x=14.485281 is a solution.
Check the solution x=-2.485281 by substituting -2.485281 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:You can use the exact solutions to factor the left side of the
original equation minus the right side of the original equation:
Since :
Since :
The product
Since and
then we could say
However the product of the first
terms of the factors does not equal
Multiply by
Let s check to see if
The factors of are and
If you would like to test yourself by working some problems similar to this
example, click on Problem
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on Contents
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