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 are perfect square trinomials. Complete the expressions to ensure equivalence between corresponding standard forms and factored forms.
|standard form||factored form|
Here the same expressions are written both in standard form and in factored form, but one or more terms are missing in each expression. To identify the missing term(s) the rules for factoring a perfect square trinomial are useful.
To help us identify the terms in the rule in the first example, we can for the expression rewrite as and as
To make the expressions match each other we will fill in where the blanks are.
We can use this reasoning to complete the other perfect square trinomials in the table. Next, we'll consider the second row.
To fill in the second blank we use that must be which gives us that When we study the first blank we can match with What is left is
This we prefer writing as Let's continue with the third row
Here the terms and match each other. Since the terms and must match each other we find that or Let's use this to fill in the blanks in the expression.
By squaring we get that the second row reads We are now going to deal with the last row in the table.
The lines match each other when giving us the first blank and the second blank Let's write them together with the blanks filled in.
We'll summarize our results by adding the found values to the table.
|standard form||factored form|
When completing the square for a quadratic expression in the form the constant must also be taken into account. When adding and subtracting the term the quadratic expression gets a constant term of
After completing its square, a quadratic expression can be rewritten into vertex form by factoring the perfect square trinomial.
Completing the square can be used when solving quadratic equations. An equation in the form is then rewritten by isolating the and terms. By adding the term to both sides of the equation, a perfect square trinomial is formed. The equation can then be rewritten by factoring the perfect square trinomial. When the equation is written on this form it can be solved with square roots.
It is easier to complete the square when the expression is written in the form Therefore, any coefficient of that does not equal should be factored out. For simplicity of the following steps, it will be assumed that For values of other than the same steps should be performed. The only difference is that the new coefficients — instead of and instead of — will be used.
The square can now be completed by adding and subtracting the constant found in Step Note that the value of the original expression will not be changed, since the result of adding and subtracting the same value is equal to The first three terms form a perfect square trinomial, which can be factored. The rest two terms do not contain the variable so their value is constant.
As it can be observed, the process of completing the square has the geometric meaning of finding a square that transforms a geometric figure compound of a square and a rectangle into another square.
Consider the quadratic function Complete the square to determine the vertex and zeros of the parabola. Then, use them to draw the graph.