A quadratic expression is a mathematical expression that involves at least one variable raised to the power of two and no variables raised to any higher powers - quadratic expressions are second degree polynomials. When only one variable has been raised to the second power, the graphical representation of the quadratic equation is always a parabola1. Despite the single graphical representation, there are three different common ways to represent parabolas symbolically in mathematical notation. The form:
... is generally preferred above the others, as the point (h,k) is the vertex of the graph and the value 'a' tells how the graph changes with respect to the x- and y-axes (that is, how wide the parabola gets, how quickly).
There are two other parabolic forms, however. The form:
is called the 'general form' of a parabola, and from an equation with this form, all the other forms are usually derived. The third form is known as 'factorised form', and is symbolically represented by:
The Point of Converting from General to Factorised Form
It is easiest to find the real zeroes2 of a parabolic equation when an equation in factorised form is set equal to zero. Because the answer is zero, one or the other (or both) of the quantities involved must equal zero. To solve for the real zeroes of a factorised expression, therefore, set each individual quantity equal to zero and solve for x.
When the equation is not given in factorised form, however, finding the real zeroes is not so easy nor so intuitive3. Therefore conversion between the two forms becomes important. This entry is concerned with the conversion of an expression in general form to factorised form4 This conversion can quickly become a time-consuming and boring task when it is attempted through trial-and-error because each term will have many distinct factors that are all possibilities for its factorised form. However, with a method of organisation (such as a table), the conversion between ax²+bx+c and (ax+b)(cx+d) becomes much easier.
The Basic Table
first term in the expression
third term in the expression
first and third terms multiplied together
'first' factor of term #1
'first' factor of term #3
the 'first' factors of terms #1 and #3, multiplied together
'second' factor of term #1
'second' factor of term #3
the 'second' factors of terms #1 and #3, multiplied together
There are a few rules which must be obeyed when filling out this table:
The factors in the first column must equal the first term of the original expression when multiplied together. (D*G=A)
The factors in the second column must equal the third term of the original expression when multiplied together. (E*H=B)
The previously-multiplied factors in the third column must themselves, when multiplied, equal the first and third terms multiplied. (F*J=C)
The previously-multiplied factors in the third column must, when added together, equal the second term in the original expression. (F+J=Second Term In Expression)
When this table has been filled out satisfactorily, the factorised form of the original expression can be derived from the table. The factorised form will have two quantities. The first quantity is the first factor of term #1 (D) added to the second factor of term #3 (H). The second quantity is the second factor of term #1 (G) added to the first factor of term #3 (E). Each of these quantities is surrounded by parentheses and written as usual - (D+H)(G+E).
The First Row
The first term in -3a2-20a-25 is -3a2. The third is -25. Multiplied together, they result in 75a2.
The Right Column
We know that the two remaining empty squares in the column furthest to the right must add up to -20 (because that is the second term in the expression -3a2-20a-25). We also know that these two new numbers must multiply to form 75a2. (It is possible to solve for these values algebraically, although it is neither necessary nor expedient to do it this way; algebraic solving involves more quadratic equations.)
The only pair of numbers that fit into these guidelines are -5a and -15a, so we put those in the right-most column. (Which value is placed where does not matter.)
The Second Row
The first box in the second row must contain a factor of -3a2, and the second box must contain a factor of -25. These two factors must form -5a when multiplied. The first number can be 1a, and the second can be -5.
|a ||-5 ||-5a |
| || ||-15a |
The Last Row
Just like those in the second row, the cells in the bottom-most row must be factors of their respective columns, and when multiplied together equal the third box of their row. What factor of -3a2, times what factor of -25, will equal -15a2? The values -3a and 5 will.
|a ||-5 ||-5a |
|-3a ||5 ||-15a |
Reading the Answer from the Table
Now that the table is complete, we can go about the relatively easy task of taking the information that it yields and presenting it in the manner we need. Diagonally down and to the right from 1a is +5. Therefore, our first quantity is (a+5). Now we start -3a and looking diagonally up and to the right: -5. Our second quantity is (-3a-5). The factorised form of the expression -3a2-20a-25 is: (a+5)(-3a-5).
When the factorised form of an expression is immediately obvious, this method is, of course, unnecessary. It acts as a guide for more difficult problems, however, and when used can greatly shorten the time spent on finding the factors and zeros of quadratic equations.
More Problems and their Solutions
6a2+13a+6 is solved by:
3x2+4x-4 is solved by:
|3x ||2 ||6x |
|x ||-2 ||-2x || (3x-2)(x+2) |
10m2-21m+8 is solved by:
|5m ||-1 ||-5m |
|2m ||-8 ||-16m || (5m-8)(2m-1) |