Note:
Solve for x in the following equation.
Problem 4.3d:
Answer: exact
answers;
and approximate answers
Solution:
The equation is already set to zero.
Rewrite the original equation as an equivalent equation without denominators by multiplying both sides by 16.
Simplify the left sides of the equation by combining like terms.
Method 1:Factoring
The equation is not easily factored, so we will not use this method.
Method 2:Completing the square
Add 173 to both sides of the equation .
Divide both sides by 128:
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 128 for a , -204 for b, and -173 for c in the quadratic formula and simplify.
Method 4:Graphing
Graph (formed by subtracting the right side of the original equation from the left side of the original equation). Graph y=0 (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 located at 2.20633314611 and -0.612583146106. This means that there are two real answers: x=2.20633314611 and
The answers are 2.20633314611 and -0.612583146106. These answers may or may not be solutions to the original equation. You must check the answers with the original equation.
Check these answers in the original equation.
Check the solution x=2.20633314611 by substituting 2.20633314611 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=-0.612583146106 by substituting -0.612583146106 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
Comment:You can use the exact solutions to factor the left side of the
For example, since , then
Since , then
Since the product
However not the only factors:
Since the first term of the product is not there must be another factor of 8:
Let s check to see whether
Therefore 8x - x- is
factored as 8
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