Perfect Squares and Completing the Square (C)

A quadratic expression of the form Az² + Bz + C, where $A\neq 0$, B and C are complex numbers can often be analysed by writing is in the form

$$Az^2 + Bz + C = A((z+a)^2 \pm b^2).$$

Doing so is called completing the square. Since we may always factor out A, we shall illustrate this in cases where A = 1. The idea is based on the obsevation that

(z+a)² = z² + 2az +a².

Notice that a is half the coefficient of z. We shall stick to examples where the coefficients are real numbers, as these occur most often, but the method is quite general.

$$z^2 + 4z + 5 = (z^2 + 2\cdot2z + 2^2) + 5-2^2 = (z+2)^2 + 1^2.$$

$$2z^2 + 12z + 4 = 2(z^2 + 2\cdot3z + 2) = 2((z^2 + 2\cdot 3z + 9) + (2-9)) = 2((z+3)^2 - (\sqrt{7})^2).$$

From this we see that the least value of 2z² + 12z + 4 as z ranges over the real numbers occurs when z = -3, and this least value is -14.

Sometimes there is more that one variable, and we can complete the square variable by variable:

$$\begin{array} {rcl} 5x^2 + 4xy + y^2 + 6x + 2y + 2 & = & y^2... ...+ 2x+1)^2 + x^2 +2x + 1\ & = & (y+2x+1)^2 + (x+1)^2\end{array}$$

so we can see that the expression 5x² + 4xy + y² + 6x + 2y + 2 is never negative, and in fact is exactly when x + 1= 0 and y + 2x + 1 = 0, that is when x=-1 and y = 1.

Exercises

1.

In each case write as a sum or difference of squares. If there is more than one variable there will be more than one answer.

(a)

x² + 12x + 2;

(b)

4x² -4x + 11;

(c)

-x² + 9x + 2;

(d)

e^2x + 4e^x + 9; (hint: treat e^x as the variable.)

(e)

x² + 4*x + 4y² + 16y + 9;

(f)

4x² + 12y² + 6x + 2y - 5;

(g)

x² + 6xy + 2y² + 10x + 3y + 5;

(h)

x² + 6xy - 2y² + 10x + 3y + 5;

2.

Find the minimum value of each of the following expressions as the variables range over all real numbers.

(a)

x² + 10x + 2;

(b)

x² - 7x -3;

(c)

x² + 4xy -2y² + 3x + 2y +3;

(d)

x² + 2y² + 3z² + 2xy + 2xz + 4yz + 2z + 10.

3.

Write as a perfect square:

(a)

$${\frac{1}{x^6} + x^6 + 2}$$;

(b)

$${\left(\frac{1}{4x^6}-x^6\right)^2 + 1}$$;

(c)

$${\left(\frac{1}{12x^3}-3x^3\right)^2 + 1}$$;

4.

Find an equivalent expression which does not involve a square root:

(a)

$$\sqrt{\left(x^{-2} - \frac{x^2}{4}\right)^2 + 1}$$

(b)

$$\sqrt{\left(2x^{-3} - \frac{x^3}{8}\right)^2 + 1}$$