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Solving Quadratic Equations

  • A quadratic equation in the variable  is and equation that can be written in the form

    Where  and  are real numbers and . When such an equation has solutions, they can be found using the quadratic formula:

    Where the notation  is shorthand for indicating two solutions: one that uses the plus sign and the other uses the minus sign.

    Ex. In the quadratic equation , we have and  Therefore, the quadratic formula yields

    Hence the two solutions are  and

  • Quadratic equations have at most two real solutions, as n the example above. However, some quadratic equations have only one real solution. For example, the quadratic equation  has only one solution, which is . In this case the expression under the square root symbol s equal to 0, and so adding or subtracting 0 yields the same result. Other quadratic equations have no real solutions; for e.g. . In this case, the expression under the square root symbol is negative, so the entire expression is not a real number.
  • Some quadratic equations can quickly be solved by factoring. For e.g. the quadratic equation  can be factored as  When a product is equal to 0, at least one of the factors must be equal to 0, which leads to two cases; either  or . Therefore,



    And the solutions are  and 2.

    Ex.  Here is another example of a quadratic equation that can easily be factored.




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