SOLVING QUADRATIC EQUATIONS


Note:

  • All methods start with setting the equation equal to zero.

    Solve for x in the following equation.




    Example 2: tex2html_wrap_inline155 tex2html_wrap_inline250

    Set the equation equal to zero by subtracting 8x and 15 from both sides of the equation.

    eqnarray16







    Method 1:tex2html_wrap_inline155Factoring

    The left side of the equation tex2html_wrap_inline254 is not easily factored, so we will not use this method.







    Method 2: tex2html_wrap_inline155Completing the square

    Add 5 to both sides of the equation tex2html_wrap_inline256

    eqnarray32

    Add tex2html_wrap_inline258 to both sides of the equation:

    eqnarray42

    Factor the left side and simplify the right side:

    eqnarray50

    Take the square root of both sides of the equation:

    eqnarray58

    Add tex2html_wrap_inline260 to both sides of the equation:

    eqnarray67

    eqnarray74

    and

    eqnarray80







    Method 3:tex2html_wrap_inline155Quadratic Formula

    The quadratic formula is tex2html_wrap_inline262

    In the equation tex2html_wrap_inline264 , a is the coefficient of the tex2html_wrap_inline266 term, b is the coefficient of the x term, and c is the constant.

    Simply insert 1 for a, -1 for b, and -5 for c in the quadratic formula and simplify.

    eqnarray101

    eqnarray106

    and

    eqnarray112







    Method 4:tex2html_wrap_inline155 Graphing

    Graph tex2html_wrap_inline274 and y=0. The graph of y=0 is nothing more than the x-axis. So what you will be looking for is where the graph of tex2html_wrap_inline274 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 2.79128784748 and -1.79128784748. This means that there are two real answers: x=2.79128784748 and -1.79128784748.

    The approximate answers are 2.79128784748 and -1.79128784748. 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=2.79128784748 by substituting 2.79128784748 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 2.79128784748 for x, then x=2.79128784748 is a solution.




    Check the solution x=-1.79128784748 by substituting -1.79128784748 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 -1.79128784748 for x, then x=-1.79128784748 is a solution.




    The solutions to the equation tex2html_wrap_inline264 are -1.79128784748 and 2.79128784748.




    Comment:tex2html_wrap_inline155 You can use the solutions to factor the original equation.

    For example, since tex2html_wrap_inline326 , then

    eqnarray153

    Since tex2html_wrap_inline332 , then

    eqnarray166

    Since the product

    eqnarray177

    then we can say that

    eqnarray188

    This means that tex2html_wrap_inline336 and tex2html_wrap_inline338 are factors of tex2html_wrap_inline340




    If you would like to work another example, click on Example

    If you would like to test yourself by working some problems similar to this example, click on Problem

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