| Before tackling the issue of how to solve a | | | | can be found in any Precalculus textbook or even |
| quadratic equation, it is important to be able to | | | | on the internet. |
| identify one! | | | | The advantage of using the formula is that it can |
| A quadratic function is any function in which the | | | | be used even when the quadratic equation is not |
| power of the leading coefficient is equal to two. | | | | factorable. Also, in the case that there are no real |
| That is, the highest power of the independent | | | | solutions for the equation, imaginary solutions can |
| variable in the function has to be two. The | | | | easily be determined. |
| standard/general form of such a function is f(x) = | | | | A minor disadvantage of using this equation is that |
| ax^2 + bx + c, where a, b, c are constants and | | | | it is possible to make an error in calculating the |
| a cannot be equal to zero. When a > 0, the | | | | solutions if a wrong number is plugged into the |
| graph of this function is a parabola that opens | | | | formula or a negative is neglected in the process |
| upward and when a < 0, the graph represents | | | | of making calculations. |
| a downward-opening parabola. | | | | A third method that can be used to solve a |
| A quadratic function may also be expressed in | | | | quadratic equation is completing the square: |
| vertex form as f(x) = a(x – h)^2 + k, where | | | | Though this method is not difficult per say, there |
| a cannot equal zero and the vertex of the graph | | | | are many instances in which computational errors |
| is at (h, k). As is the case with the standard form | | | | can be made. The good thing about completing |
| of the quadratic function, if a > 0, the graph | | | | the square is that is can also be used to convert |
| opens upward and if a < 0, the graph opens | | | | a quadratic function from standard or general |
| downward. | | | | form to vertex form. |
| Many times, the quadratic function is set equal to | | | | Keep in mind that there are many advantages to |
| zero (with f(x) = 0) in order to solve for the | | | | having a quadratic equation in vertex form. |
| x-intercepts (or roots) of the function. This | | | | The fourth method used to solve a quadratic |
| process may also be referred to as solving a | | | | equation is the graphing calculator method: |
| quadratic equation. When this is done, the | | | | The original quadratic function can be entered into |
| quadratic function in standard form becomes a | | | | the calculator either in standard or vertex form. |
| quadratic equation of the form ax^2 + bx + c = | | | | Once this is done, the calculator's calculation |
| 0. | | | | capabilities can be used to find the roots of the |
| The first method that can be used to solve a | | | | function. |
| quadratic equation is factoring: | | | | On most of the TI calculators, the quadratic |
| The quadratic equation in standard form can be | | | | function can be entered once the "Y =" button is |
| factored into two binomials, that is, two | | | | pressed. After that, the graphing window may |
| polynomials with two terms. For example, 2x^2 + | | | | have to be adjusted in order to see the complete |
| x – 3 = 0 can be factored into (2x + 3)(x | | | | graph. |
| – 1) = 0. | | | | The next step would involve pressing the "2nd" |
| Once this is done, each of these binomials can be | | | | and "TRACE" buttons simultaneously in order to |
| solved for x. These values of x represent the | | | | access the CALC menu. Under the CALC menu, |
| x-intercepts. | | | | the "root" option should be selected in order to |
| One drawback to this method is that not every | | | | determine the roots. |
| quadratic equation can be factored. Also, some | | | | Note that each root has to be determined |
| factorable quadratic equations are not very easy | | | | separately. |
| to factor. | | | | In conclusion, there are four options that can be |
| 2. The second method that can be | | | | used to solve a quadratic equation: factoring, the |
| used is the quadratic formula: | | | | quadratic formula, completing the square, and the |
| The equation must be converted to standard | | | | graphing calculator method. They should each be |
| form in order to use the formula. This formula | | | | known so that they can be used interchangeably. |