SOLVING QUADRATIC EQUATIONS

Note:

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





    Solve for x in the following equation.

    Example 3: tex2html_wrap_inline155 tex2html_wrap_inline193

    Set the equation equal to zero by subtracting 60 and adding 2x to both sides of the equation.

    eqnarray17





    Method 1: tex2html_wrap_inline155 Factoring


    eqnarray25


    eqnarray28





    Method 2: tex2html_wrap_inline155 Completing the square

    Add 78 to both sides of the equation.

    eqnarray36

    Add tex2html_wrap_inline199 to both sides of the equation:

    eqnarray47

    Factor the left side and simplify the right side:

    eqnarray55

    Take the square root of both sides of the equation :

    eqnarray63

    Subtract tex2html_wrap_inline201 to both sides of the equation :

    eqnarray78

    eqnarray85

    and

    eqnarray93





    Method 3: tex2html_wrap_inline155 Quadratic Formula

    The quadratic formula is tex2html_wrap_inline203

    In the equation tex2html_wrap_inline205 , a is the coefficient of the tex2html_wrap_inline207 term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, +7 for b, and -78 for c in the quadratic formula and simplify.

    eqnarray114

    eqnarray119

    eqnarray126

    and

    eqnarray132





    Method 4: tex2html_wrap_inline155 Graphing

    Graph y= the left side of the equation or tex2html_wrap_inline211 and graph y= the right side of the equation or 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_inline211 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 -13 and 6. This means that there are two real answers: x=-13 and tex2html_wrap_inline229 Check these answers in the original equation.





    Check the solution x=-13 by substituting -13 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 -13 for x, then x=-13 is a solution.



    Check the solution x=6 by substituting 6 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 6 for x, then x=6 is a solution.



    The solutions to the equation tex2html_wrap_inline193 are -13 and 6.



    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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    Author:Nancy Marcus

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