One of the most useful techniques in algebra is that of completing the square. The name is appropriate as the geometric interpretation encompasses the formation of a square from a rectangle by the addition of an appropriate quantity. Geometry aside, this technique has many applications, not only in algebra, but also in more advanced realms such as integration, which is a key component of integral calculus. Here we will see that this technique can be had rather inexpensively.
Completing the square involves taking a non-perfect square trinomial and converting it into a perfect square. Actually, this technique is performed when you have a quadratic equation set to zero, as in x^2 + 10x - 5 = 0. If you recall, a perfect square trinomial is one in which the middle coefficient is equal to twice the product of the square roots of both the leading coefficient and the constant term. What a mouthful! Let's look at a specific example. Take the quadratic trinomial x^2 + 10x + 25. The leading coefficient is 1, the number (which is understood) in front of the x^2 term. The middle coefficient is 10, and the constant term is 25. The square root of 1 is naturally 1; the square root of 25 is 5; 2*1*5 is 10, which is the middle coefficient. Thus x^2 + 10x + 25 qualifies as a perfect square trinomial.
So what is so special about these trinomials? Well for one, they can always be factored into the form (x +/- c)^2. In other words, we can always factor them as (x + c)^2 or (x - c)^2, where c is a constant and the "+" or "-" is dictated by the sign of the middle coefficient. Once factored, we can easily solve any quadratic equation by performing the simple operation of taking the square root and adding or subtracting the constant c. To see this, let us look at a specific example.
Suppose we wish to solve the quadratic equation x^2 + 8x - 10 = 0. You cannot solve this by factoring. You can of course go directly to the quadratic formula, but an even quicker way is to complete the square, and this is how we shall do it. Isolate the x-terms, namely x^2 and 8x, on one side of the equation and bring the constant term to the other. Remember that when we move the -10 over we get +10. Thus we have x^2 + 8x = 10. Now begin the process of converting x^2 + 8x into a perfect square. We take half of 8, which is 4 and square it to get 16. We add this quantity to both sides of the equation to get x^2 + 8x + 16 = 10 + 16 = 26. Now if you check the conditions which make a trinomial perfect, you will see that x^2 + 8x + 16 fits the bill. That is 2*4*1 = 8.
Since the trinomial is now perfect, we can factor it into (x + 4)^2, that is we take the x term, half of 8, and the "+" sign, since the middle term is positive. We write (x + 4)^2 = 26. To solve this equation, we simply take the square root of both sides, remembering to take the "+" and "-" part. (Remember: when we take a square root in an equation, we always consider both the positive and negative values). Thus we have (x + 4) = +/- the square root of 26. (Since I cannot use the square root symbol in this article, I will write 26^.5 as the square root of 26; actually this is true since the square root is the one-half power.) To finish this off, we subtract the 4 from both sides to solve for x, and we get x = -4 +/- (26)^.5, that isx = -4 + (26)^.5 or x = -4 - (26)^.5. Since (26)^.5 is equal to a little more than 5, about 5.1, we have that x is equal to about 1.1 or -9.1.
With this technique, you can now solve any quadratic, regardless of whether it is factorable or not, without resorting to the quadratic formula. To sum up, all you need do is the following (As you read these steps, refer back to the example just done):
1) Isolate the x terms on one side of the equation and the constant term on the other;
2) Take half the middle coefficient, square it and add it to both sides of the equation;
3) Factor the trinomial using (x +/- c)^2, where c is equal to half the middle term, and the sign is taken according to the sign of the middle coefficient; and
4) Take the square root of both sides, remembering to consider the +/- cases, and add or subtract c to both sides.
With the ammunition given above, you are now expert at completing the square and solving any quadratic equation. Isn't life grand!
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