Note:
Solve for x in the following equation.
Example 1:
The equation is already set to zero.
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 2.
Method 1: Factoring
The equation is not easily factored. Therefore, we will not use this method.
Method 2: Completing the square
Add 10 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 16 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 1 for a, -32 for b, and -10 for c in the quadratic formula and simplify.
Method 4: Graphing
Graph the left side of the equation, and graph the right side of the equation, The graph of is nothing more than the x-axis. So 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 32.309506 and one at -0.309506.
The answers are 32.309506 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=32.309506 by substituting 32.309506 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=-0.309506 by substituting -0.309506 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 32.309506 and - 0.309506.
Comment: You can use the exact solutions to factor the original equation.
Since
Since
The product
Since
then we could say
However the product of the first terms of the factors does not equal
Multiply
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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