Algebra Know-It-ALL

374 Quadratic Equations with Real Roots

x^2 + 4 x+ 4 = 16

9 x^2 + 12 x+ 4 = 25

Morphing into a perfect square

When we come across a quadratic equation in polynomial standard form, we can sometimes

add a positive real number to both sides, getting a perfect square on the left side. That will

leave us with a nonzero value on the right, but as long as the left side is a perfect square, we

can solve the equation just as we did in the four cases above. We can take the first quadratic

from the above list and subtract 1 from each side, getting

x^2 + 2 x= 0

We can take the second equation and subtract 4 from each side, getting

x^2 − 2 x− 3 = 0

In the third equation, we can subtract 16 from each side to obtain

x^2 + 4 x− 12 = 0

Finally, in the fourth equation, we can take 25 away from each side and get

9 x^2 + 12 x− 21 = 0

If you add the right constants to both sides of each of these equations, you’ll get quadratics

with perfect squares on their left-hand sides.

Are you confused?

Now that we’ve taken four solutions and manufactured four problems from them, let’s retrace our steps

and get the solutions back. In this way, we can get a good “feel” for how completing the square actually

works. Imagine that we’re confronted with the following four quadratics in polynomial standard form:

x^2 + 2 x= 0

x^2 − 2 x− 3 = 0

x^2 + 4 x− 12 = 0

9 x^2 + 12 x− 21 = 0

We can take the first of these equations and add 1 to each side, getting

x^2 + 2 x+ 1 = 1

That gives us a perfect square on the left side. (Recognizing perfect squares when they appear in polynomial

form is a “sixth sense” that evolves over time, and it takes practice to develop it.) Factoring, we obtain

(x+ 1)^2 = 1