Note:
Solve for x in the following equation
Example 2:
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 12. All the denominators of the
original equation divide into 12 evenly.
Set the equation equal to zero by subtracting 2x from both sides of the equation:
Method 1: Factoring
The equation is not easily factored. Therefore, we will not use this method.
Method 2: Completing the square
Divide both sides of the equation by 9.
Subtract from 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 :
Add to 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
9.48527114763 and one at 0.07028440793.
The answers are 9.48527114763 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=9.48527114763 by substituting 9.48527114763 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 x=0.07028440793 by substituting 0.07028440793 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 9.48527114763 and 0.07028440793.
Comment: You can use the exact solutions to factor the left side of the
original equation set to zero:.
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
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