The quadratic formula is a formula which will help you to solve any quadratic equation, even if it does not factor "nicely."

In this brief unit, we will explore the concepts that lead up to this remarkable formula. The quadratic formula begins with the concept of completing a square. So let's review what a quadratic square is.

If you begin with the binomial (2x + 3), and square it, you will get the following:

(2x + 3)² = 4x² + 12x + 9

If you look at the numbers on the left, you might notice some relationships to the numbers on the right. For example, 4 is the square of 2, and 9 is the square of 3. Is this a coincidence? Let's try another binomial, and find out:

(7x - 5)² = 49x² - 70x + 25

Interesting! We have 7, and its square (49), and we also have 5 and its square (25). But what about that middle number, the 70? Well, it turns out that -70 is twice the product of 7 and -5.

So if we're asked to square a binomial, all we have to do is square the coefficient of x, square the constant term, find twice the product of the coefficient and the constant, and then we just put it together.

Example #1
Find the square of (2x + 6).

Solution
2² = 4
6² = 36
2 x 2 x 6 = 24

Thus, (2x + 6)² = 4x² + 24x + 36

Example #2
Find the square of (8x - 9).

Solution
8² = 64
9² = 81
2 x 8 x (-9) = 144

Thus, (8x - 9)² = 64x² + 144x + 81

If you're the kind of student that likes to see why a rule works, it's easy to show. Instead of using actual numbers, we'll use a to represent the coefficient of x, and b to represent the constant. So let's find the square of (ax + b).

(ax + b)² = 
(ax + b)(ax + b) =
ax(ax + b) + b(ax + b) =
a²x² + axb +bax + b² = 
a²x² + 2abx + b²

And there you have it. Square the a, square the b, and take twice the product of the two for the middle term!

For each question, write the squared binomial as a quadratic.