To be able to rewrite an expression to include a perfect square trinomial, it is first necessary to be able to recognize them. This can be achieved by expanding the square of a general binomial. The factoring is then done in the opposite direction. There are two kinds of perfect square trinomials, leading to different signs between the terms in the binomial.
If a trinomial is in the form where and are variables or positive numbers, it is a perfect square trinomial that can be factored as This is shown by expanding the squared binomial.
Thus, the trinomial and the square are equal.
If a trinomial instead is in the form it is also a perfect square trinomial and can be factored as This is shown by expanding the squared binomial.
The trinomial and the square are indeed equal.
The expressions in the table below are perfect square trinomials. Complete the expressions to ensure equivalence between corresponding standard forms and factored forms.
|standard form||factored form|
To begin, we'll write the expression in the first row in factored form without explicitly factoring it. If factors to be __ both of the following statements must be true. It follows that Thus, in factored form We can use this reasoning to complete the other perfect square trinomials in the table. Next, we'll consider the second row. Notice that Thus, Moving on to the third row, we can see that we're given the coefficient of the -term. Since must equal The constant term of the trinomial can be found by squaring Thus, In the last row of the table, we're given the constant term of the trinomial. Since must equal it follows that The coefficient of the -term can be found by adding to itself. Thus, We'll summarize our results by adding the found values to the table.
|standard form||factored form|
For an expression of the form where is some number, a constant can be added to create a perfect square trinomial. This process is called completing the square. If the value of is chosen with care,is a perfect square trinomial and can be factored. Completing the square of a quadratic function in standard form turns it into its vertex form. It can also be used to solve any quadratic equation.
All quadratic expressions in standard form can be rewritten by completing the square. One example of how this can be used in practice is to rewrite the function in vertex form to determine the vertex.
Completing the square is easiest done when the expression is written in the form Therefore, any coefficient of that does not equal should be factored out. Here, this means factoring out of the expression.
The constant that is necessary to complete the square can now be identified by focusing on the - and - terms, while ignoring the rest. It's found either by comparing the binomial to the general perfect square trinomials, or by squaring half the coefficient of The binomial that should be studied in is Since the terms are added, it can be compared to the perfect square trinomial The second term of the trinomial consists of the factors and Rewriting so that its second term consists of three factors, being the first, helps with identifying how to complete the square.
It can now be identified that corresponds to in the trinomial, and that corresponds to Therefore, the missing constant is Alternatively, it can be found by squaring half the coefficient of in this case. With both methods, the missing constant is found to be Leaving the constant as a power makes the next steps a bit quicker.
The square can now be completed by adding and subtracting the found constant. In the example, the constant should be added and subtracted inside the parenthesis. The resulting trinomial, is now a perfect square.
The perfect square trinomial can now be factored as normal.
Consider the quadratic function Complete the square to determine the vertex and zeros of the parabola. Then, use them to draw the graph.