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_inline155

    The equation is already equal to zero.





    Method 1: tex2html_wrap_inline155 Factoring

    The equation can be written in the equivalent form of tex2html_wrap_inline157 The only way that a product can equal zero is if one or both of the factors equal zero.

    eqnarray44

    The answers are tex2html_wrap_inline159





    Method 2: tex2html_wrap_inline155 Completing the square

    Subtract 5 from both sides of the equation.

    eqnarray51

    Add tex2html_wrap_inline161 to both sides of the equation.

    eqnarray60

    Factor the left side and simplify the right side.

    eqnarray64

    Take the square root of both sides of the equation,

    eqnarray68

    Add 3 to both sides of the equation.

    eqnarray71

    and

    eqnarray73





    Method 3: tex2html_wrap_inline155 Quadratic Formula

    The quadratic formula is tex2html_wrap_inline163

    In the equation tex2html_wrap_inline155 , a is the coefficient of the tex2html_wrap_inline167 term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, -6 for b, and 5 for c in the quadratic formula and simplify

    .

    eqnarray90

    and

    eqnarray100





    Method 4: tex2html_wrap_inline155 Graphing

    Graph y= the left side of the equation or tex2html_wrap_inline179 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_inline179 crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation. The x-intercepts are 1 and tex2html_wrap_inline193 The answers are 1 and tex2html_wrap_inline201





    Check these answers in the original equation.

    Check the answer x=1 by substituting 1 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 for x, then x=1 is a solution.

    Check the solution x=5 by substituting 5 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 5 for x, then x=5 is a solution.




    If you would like to go 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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