Kevin B. answered • 10/25/22
A Specialist in Math and Physics
Answer
A zero of a function just means "the x-value that makes the function equal 0". That means we just set the function equal to 0 and solve for x:
4x2 + 16x + 9 = 0
But x is not easy to solve in quadratic functions. They must be "factored" first to get them into a solvable form. We are instructed to "complete the square", which just means instead of factoring it into something like this:
(nx - a)(mx - b) = 0
we instead need to factor it like this:
1: (nx - p)2 + m = 0
meaning we need to figure out which m, n, and p values would be equivalent to our function. Here's how. It will be easier if we change the form of our function to look like this:
2: n2x2 -2npx + (p2 + m) = 0
Equation 1 is just what you get when you perform a FOIL/double-distribute on equation 1. So if we can figure out what to use for m, n, and p, we can get our function into the same form as 1. It looks weird, but it's easy to do. Our function was
4x2 + 16x + 9 = 0
so that just means the n2 part is 4, the -2np part was 8, and the (p2 + m) part is 9. Write it all down:
n2 = 4
-2np = 16
p2 + m = 9
Now we solve for n, m, and p:
n2 = 4
n = √4 = 2 --(we can neglect the negative square root)
-2np = 16
-2(2)p = 16
p = -4
p2 + m = 9
(-4)2 + m = 9
m = -7
All we have done is find the values we can use to change the form of our function to a convenient one. So plug them in:
n2x2 -2npx + (p2 + m) = 0
(2)2x2 - 2(2)(-4)x + ((-4)2 + (-7)) = 0
We know we did it right, because if we simplify everything, it just goes back to
4x2 + 16x + 9 = 0
so we have not changed any values, just changed to form. No math rules have been broken.
It can be changed directly to the form of equation 1:
(nx - p)2 + m = 0
(2x - (-4))2 + (-7) = 0
(2x + 4)2 - 7 = 0
In this form, x is easily solvable!
(2x + 4)2 - 7 = 0
add 7 to both sides
(2x + 4)2 = 7
take the square root of both sides
2x + 4 = ±√7
subtract 4 from both sides
2x = ±√7 - 4
divide both sides by 2
x = (±√7 - 4) / 2
This is our solution. But it's more useful if we expand the ± and show the two distinct values for x:
x = (±√7 - 4) / 2
x = (+√7 - 4) / 2, (-√7 - 4) / 2
x = (1/2)√7 - 2, -(1/2)√7 - 2
x = (1/2)√7 - 2, -(1/2)√7 - 2
This answer is exact, so it's best to leave it this way. If we change the square roots into decimals, it will only be the approximate value.
Since we were instructed to check our answer, we should plug these values into the original function. If these really are the zeroes of the quadratic function, plugging them in will each give a result of 0 (because that's the definition of a zero).
f(x) = 4x2 + 16x + 9
f((1/2)√7 - 2) = 4[(1/2)√7 - 2]2 + 16[(1/2)√7 - 2] + 9
=4[(1/4)(7) - 2√7 + 4] + 8√7 - 32 + 9
=7 - 8√7 + 16 + 8√7 - 23 = 0
Good!
f(x) = 4x2 + 16x + 9
f(-(1/2)√7 - 2) = 4[-(1/2)√7 - 2]2 + 16[-(1/2)√7 - 2] + 9
= 4[(1/4)(7) + 2√7 + 4] - 8√7 - 32 + 9
= 7 + 8√7 + 16 - 8√7 - 23 = 0
Good!
We have found the correct answer.
Extra Notes: What is Completing the Square?
As I mentioned above, completing the square is just a fancy way to factor a quadratic expression which allows for easy solving of the zeroes. You have probably learned a variety of different methods for determining or approximating the zeroes of a quadratic function (graphing, different types of factoring, guess-and-check), but completing the square is the only basic method that finds the exact answer for ANY quadratic, even if there are imaginary/complex roots.
Here's the justification. If
1: f(x) = (nx - p)2 + m
Then its zeroes are always easily solvable by doing this:
(nx - p)2 + m = 0
(nx - p)2 = -m
nx - p = ±√(-m)
x = [±√(-m) + p] / n
The main problem is that quadratic functions are usually given in the form
3: f(x) = ax2 + bx + c
and a general method for factoring 3 into the form of 1 is not obvious. But if we start with 1 and work backwards to get it to the expanded form in 3:
f(x) = (nx - p)2 + m
= (nx - p)(nx - p) + m --expand the exponent
= [n2x2 - npx - npx + p2] + m --FOIL/double-distribute
= n2x2 - 2npx + p2 + m
But p2 and m are both just constants, so we can "wrap them together" with parentheses to show that that are added together to represent the constant part of the quadratic:
4: f(x) = n2x2 - 2npx + (p2 + m)
which is the same form as
f(x) = ax2 + bx + c
The whole point of representing it like in 4 is that all of the same m, n, and p variables that are used in the form of 1 are intact, so we can figure out what they are and rewrite our function in a solvable form.
Since a is the same coefficient as n2 and b is the same coefficient as -2np and c is the same constant as (p2 + m), so we can equate them as such to determine the values of m, n, and p:
a = n2
b = -2np
c = p2 + m
Since a, b, and c are all known values, this is a system of 3 equations in 3 variables, and so all 3 variables can be solved. Afterward, we just plug them all into
(nx - p)2 + m = 0
And solve for x.
"Wait! This isn't the ONLY method that works for all quadratics. What about the quadratic formula?"
If you are thinking this, you should be proud. It kind of sounds like I was lying before if you already know that you can use the quadratic formula on all quadratic functions instead of completing the square. You are right, but you should know that using the quadratic formula is the same thing as completing the square. Completing the square for a quadratic that is still in the form
ax2 + bx + c = 0
is actually where the quadratic formula comes from!
