We can solve an equation of the form
x2+bx=c, where
b and
c are , by . Notice that we can always rewrite any in this form by isolating the terms and dividing by the of
x2 if needed. Let's see an example.
4x2+24x+32=0
4x2+24x=-32
x2+6x=-8
Now we need to ensure that we have an expression that is a . Then, we will be able to it as the and take the square root on both sides to find the solutions. Let's recall the expansion of the square of a binomial.
(x+a)2=x2+2ax+a2
By adding the appropriate constant, we can obtain a . Let's compare the right-hand side of this formula to the left-hand side of our expression.
x2+2ax+a2x2+bx+?
Notice that the
b parameter of our expression corresponds to the factor
2a in the expansion for the square of the binomial. We can find the quantity corresponding to
a2 that we would have to add from this relationship by solving for
a first and then squaring both sides.
This means that by adding the term
(2b)2 we can get a perfect square trinomial! Let's try this with the example equation we had at the beginning,
x2+6x=-8. Notice that here
b=6.
Then, adding
9 to both sides of
x2+6x=-8 will let us rewrite the left-hand side as the square of a binomial.
x2+6x=-8
x2+6x+9=1
(x+3)2=1
x+3=±1
x=-3±1
As we can see, the solutions for this equation are
x1=-3+1=-2 and
x2=-3−1=-4.
Summary of the Procedure
As we saw above, we can complete the square by following the next steps.
- We isolate the terms containing the variable x.
- We divide both sides by the coefficient of x2.
- We identify the coefficient of the linear term as b.
- We add (2b)2 to both sides.