quadex 1

Perfect square
on the left side
(and right side in this problem) .
When solving an equation involving a perfect square (such as x2 = 49 seen at the left), taking the square root of both sides of the equation quickly yields the result.

This same square rooting process works nicely on equations with the square of a binomial on the left side (such as (x + 1)2 = 9 as seen at the right). We could even replace (x + 1)2 with x2 + 2x + 1, and start with a perfect square trinomial on the left side.
quadex2

statement
Creating a perfect square trinomial on the left side of a quadratic equation, with a constant (number) on the right, is the basis of a method called completing the square.

In plain English, we are going to "force" a perfect square trinomial
on the left hand side of the equation to help us find the solution more quickly.

bullet To solve quadratic equations by Completing the Square:

1. Check to see if the leading coefficient is one. If not, divide each term by the leading coefficient.
x2 + 4x - 2 = 0
Leading coefficient is one.
2. If there is a constant term on the left side of the equation, move the constant term to the right side.
x2 + 4x = 2
3. Set up the problem to receive the addition of the value which will create a perfect square trinomial on the left side. Inserting boxes may remind you to add the value to BOTH sides of the equation.
boxa
4. To get the needed value for creating a perfect square trinomial, take half of the coefficient of the middle term (x-term) and square it. Add this value to both sides of the equation (put this value in the boxes).

halfmiddle
boxfill

(Be sure to take note of the "sign" of half the coefficient of the middle term, as it will be used when factoring the perfect square trinomial.
In this case, +2.)
5. Factor the perfect square trinomial on the left side.
(x + 2)2 = 6
6. Now that there is a perfect square on the left side, take the square root of both sides. Solve for x. Be sure to remember to use "plus and minus" to arrive at the two roots of the equation.

Check your solutions in the original equation to see that they work.
check1
Always assume that answers are to be left in "exact" form (not rounded), unless told otherwise..

boxend
In real-world problems and for graphing, these values may be expressed as rounded decimal values:
x ≈ 0.4494897428
x ≈ -4.449489743

Let's verify, geometrically, that adding the square of half of the coefficient
of the x-term actually creates a perfect square trinomial.

cspainting
To solve x2 + bx + c = 0 by completing the square, we first move the constant, c, to the right side, x2 + bx = -c .

We then create a perfect square trinomial on the left by adding the square of half the coefficient of the x-term to both sides. That is, we add add termsto both sides. Let's look at just the left side of that equation, and "paint a picture" of the perfect square trinomial.
paint1
paingpic1
As seen in the picture, the area of the square can also be expressed as the sum of the areas of its parts: paintpic3
So both expressions are equivalent, verifying that this process creates a perfect square trinomial.

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hintgal
It will prove worthwhile to know this process as there will be other applications of completing the square. For example, completing the square will be used to derive important formulas, to create new forms of quadratics, and to discover information about conic sections (parabolas, circles, ellipses and hyperbolas).

See Completing the Square Examples with worked out steps
on various types of problems.

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