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Quadratic Equations

Completing the Square

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Quadratic expressions, functions, and equations can all be manipulated in useful ways with a method called completing the square. This method is based on the concept of perfect square trinomials.

Concept

Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. There are two general types of perfect square trinomials that can be useful when dealing with quadratic functions or quadratic equations.

Perfect Square Trinomial Square of Binomial
a2+2ab+b2 (a+b)2
a22ab+b2 (ab)2
A perfect square trinomial should have one of the forms shown in the table — that is, the first and the last terms are perfect squares and the middle term is two times the product of the square roots of the first and last terms.
Once a quadratic expression is recognized as a perfect square trinomial, it can be written as a square of a binomial by factoring the perfect square trinomial.

Method

Factoring a Perfect Square Trinomial

For a trinomial to be factorable as a perfect square trinomial, the first and last terms must be perfect squares and the middle term must be two times the square roots of the first and last terms. Consider the following expression.
To factor this trinomial, there are three steps.
1
Confirm That the First and Last Terms Are Perfect Squares
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One good way to recognize if a trinomial is a perfect square trinomial is to look at its first and last terms. If they are both perfect squares, there is a good chance that it is a perfect square trinomial. In the given expression, the first and last terms can be written as the squares of 4x and 11, respectively.
These perfect squares show that the expression could be a perfect square trinomial. However, this is not enough to decide yet.
2
Confirm That the Middle Term Is Twice the Product of the Square Roots of First and Last Terms
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The next step is to check whether the middle term is two times the square roots of the first and last terms.
It can be seen that the given expression satisfies this condition as well.
3
Write as a Square of a Binomial
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Since the expression satisfies both conditions, it is a perfect square trinomial. Therefore, it can be written as a square of a binomial where 4x and 11 are the first and second terms of the binomial, respectively.

Example

Create the perfect square trinomial

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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
x22x+1 (x__ ___)2
x2___x+___ (x7)2
x210x+___ (x__ ___)2
x2+___x+100 (x__ ___)2
Show Solution expand_more

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.

a2+2ab+b2=(a+b)2
and
a22ab+b2=(ab)2

To help us identify the terms in the rule in the first example, we can for the expression x22x+1 rewrite 1 as 12 and 2x as 2x1.


x22x1+12=(x__ ___)2

To make the expressions match each other we will fill in 1 where the blanks are.


x22x1+12=(x1)2

We can use this reasoning to complete the other perfect square trinomials in the table. Next, we'll consider the second row.

a22ab+b2=(ab)2
x2___x+___ =(x7)2

To fill in the second blank we use that b must be 7, which gives us that b2=72. When we study the first blank we can match a with x. What is left is 2b=27.


x22x7+72=(x7)2

This we prefer writing as x214x+49=(x7)2. Let's continue with the third row


x210x+___=(x__ ___)2

Here the terms a and x match each other. Since the terms 2ab and 10x must match each other we find that 2b=10, or b=5. Let's use this to fill in the blanks in the expression.


x210x+52=(x5)2

By squaring 5 we get that the second row reads x210x+25=(x5)2. We are now going to deal with the last row in the table.


x2+___x+100=(x__ ___)2

The lines match each other when b=10, giving us the first blank 210=20 and the second blank +10. Let's write them together with the blanks filled in.


x2+20x+100=(x+10)2

We'll summarize our results by adding the found values to the table.

standard form factored form
x22x+1 (x1)2
x214x+49 (x7)2
x210x+25 (x5)2
x2+20x+100 (x+10)2

Concept

Completing the Square

Completing the square is a method by which a quadratic expression is rewritten as a difference of a perfect square trinomial and a constant. Commonly this is done by adding and subtracting a constant, p.
By choosing the constant, p, as the quadratic expression becomes a difference of a perfect square trinomial and a constant.
The quadratic expression can then be factored and written and written in vertex form.

Extra

Completing the Square, In-Depth

Concept

Quadratic Expressions in the Form x2+bx+c

When completing the square for a quadratic expression in the form x2+bx+c, the constant c must also be taken into account. When adding and subtracting the term the quadratic expression gets a constant term of

Concept

Factoring

After completing its square, a quadratic expression can be rewritten into vertex form by factoring the perfect square trinomial.

Concept

Solving Quadratic Equations

Completing the square can be used when solving quadratic equations. An equation in the form x2+bx+c=0 is then rewritten by isolating the x2- and x-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.

Concept

Geometric Interpretation of Completing the Square

It can be helpful to visualize what's going on when completing the square. Press the button below to see an animation of this concept.
Complete the square

Method

Completing the Square

In a perfect square trinomial, there is a relationship between the coefficient of the x-term and the constant term — the constant term is equal to the square of half the coefficient of the x-term.
This relationship can be used to form a perfect square trinomial by adding a constant c to any expression in the form x2+bx.
The process of finding the constant c can be visualized by using algebraic tiles. Consider the following expression.
The expression is represented using algebraic tiles. Then, a square is created by rearranging the existing tiles and adding more tiles. The following applet summarizes this process.
Showing the process of completing the square by using algebraic tiles
This process is called completing the square. To complete the square for an expression algebraically, these steps can be followed.
1
Identify the Coefficient of the x-Term
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For the given expression, the value of b is 6.
2
Calculate
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Once the value of b is identified, calculate the square of half of the value of b.
32
9
3
Add to the Initial Expression
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Add to the expression to obtain a perfect square trinomial.
In this case, 9 should be added to x2+6x.
4
Factor the Perfect Square Trinomial
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The expression obtained in the previous step can be now factored as the square of a binomial.
This will be applied to the expression x2+6x+9.
x2+6x+9
x2+23x+9
x2+2x3+9
(x+3)2
Therefore, a perfect square trinomial is obtained by adding a constant to the initial expression in the form x2+bx.

Example

Solve the quadratic equation by completing the square

fullscreen
Consider the quadratic function
f(x)=x2x0.75.
Complete the square to determine the vertex and zeros of the parabola. Then, use them to draw the graph.
Show Solution expand_more
To begin, we'll complete the square on f by focusing on the coefficient of x, which is -1. Since 0.25 is the constant term that will create a perfect square trinomial.
f(x)=x2x0.75
f(x)=(x0.5)20.750.25
f(x)=(x0.5)21
Notice that f is now written in vertex form. It can be seen that f's vertex is (0.5,-1). To find the zeros we can set f(x) equal to 0 and solve.
0=(x0.5)21
1=(x0.5)2
(x0.5)2=1
x0.5=±1
x=0.5±1
Thus, the zeros of the parabola are x=-0.5 and x=1.5. Lastly, we can use the vertex and the zeros to draw the graph of f.
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