We can take the square root of both sides and get
x+ 1 =± 1
which can be expressed as the pair
x+ 1 = 1 or x+ 1 =− 1
The solutions are found to be x= 0 or x=−2, so the solution set is {0,−2}. The other three equations can
be worked out in similar fashion.
Are you still confused?
Do you wonder what happens if, in order to complete the square in a quadratic, you must subtract a posi-
tive number from both sides, getting a negative number on the right side? That’s a good question. In that
case, the roots turn out to be imaginary or complex. We’ll deal with such equations in Chap. 23.
Here’s a challenge!
Go through maneuvers similar to those we just completed, but with the second, third, and fourth quadrat-
ics from above:
x^2 − 2 x− 3 = 0
x^2 + 4 x− 12 = 0
9 x^2 + 12 x− 21 = 0
Solution
You’re on your own! Start with perfect squares on the left sides of the equals signs and positive numbers on
the right, and then take away those positive numbers from both sides to “unsquare” the equations.
The Quadratic Formula
The technique of completing the square can be applied to the general polynomial standard
form of a quadratic equation. This gives us a tool for solving quadratics by “brute force”: the
so-calledquadratic formula.
Deriving the formula
Remember the polynomial standard form where x is the variable, and a,b, and c are real-
number constants with a≠ 0. The general formula is
ax^2 +bx+c= 0
Let’s rewrite this as
ax^2 +bx=−c
The Quadratic Formula 375
