Completing the Square (Shortcut)

Completing the Square is where we take a Quadratic Equation

like this ax^2+ bx + c = 0 and turn it into this a(x + d)^2 + e = 0





For those of you in a hurry, I can tell you that:

d = b / 2a

e = c − [ (b^2) / 4a ]


Too short for you to even understand? Let's look at another shortcut but not so short method.

Here is a quick way to get an answer. You may like this method.


First think about the result we want: (x+d)^2 + e

After expanding (x+d)^2 + e we get: x^2 + 2dx + d^2 + e


Now see if we can turn our example into that form to discover d and e


Example:

try to fit x^2 + 6x + 7 into x^2 + 2dx + d^2 + e

Now we can "force" an answer:

  • We know that 6x must end up as 2dx, so d must be 3

  • Next, we see that 7 must become d^2 + e = 9 + e, so e must be −2

And we get the answer (x+3)^2 − 2



Solving General Quadratic Equations by Completing the Square


We can complete the square to solve a Quadratic Equation (find where it is equal to zero).

But a general Quadratic Equation can have a coefficient of a in front of x^2:

ax^2 + bx + c = 0


But that is easy to deal with ... just divide the whole equation by "a" first, then carry on:

x^2 + (b/a)x + c/a = 0


Steps

Now we can solve a Quadratic Equation in 5 steps:


Step 1: Divide all terms by a (the coefficient of x^2).

Step 2: Move the number term (c/a) to the right side of the equation.

Step 3: Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.


We now have something that looks like (x + p)^2 = q, which can be solved rather easily:

Step 4: Take the square root on both sides of the equation.

Step 5: Subtract the number that remains on the left side of the equation to find x.



Examples


Example 1

Solve x^2 + 4x + 1 = 0


Step 1 can be skipped in this example since the coefficient of x^2 is 1


Step 2 Move the number term to the right side of the equation:

x^2 + 4x = -1


Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation.


(b/2)^2 = (4/2)^2 = 2^2 = 4

x^2 + 4x + 4 = -1 + 4

(x + 2)^2 = 3


Step 4 Take the square root on both sides of the equation:

x + 2 = ±√3 = ±1.73 (to 2 decimals)


Step 5 Subtract 2 from both sides:

x = ±1.73 – 2 = -3.73 or -0.27


And here is an interesting and useful thing.

At the end of step 3 we had the equation: (x + 2)^2 = 3

It gives us the vertex (turning point) of x^2 + 4x + 1: (-2, -3)



Example 2

Solve 5x^2 – 4x – 2 = 0


Step 1 Divide all terms by 5

x^2 – 0.8x – 0.4 = 0


Step 2 Move the number term to the right side of the equation:

x^2 – 0.8x = 0.4


Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation:

(b/2)^2 = (0.8/2)^2 = 0.4^2 = 0.16

x^2 – 0.8x + 0.16 = 0.4 + 0.16

(x – 0.4)^2 = 0.56


Step 4 Take the square root on both sides of the equation:

x – 0.4 = ±√0.56 = ±0.748 (to 3 decimals)


Step 5 Subtract (-0.4) from both sides (in other words, add 0.4):

x = ±0.748 + 0.4 = -0.348 or 1.148



Why "Complete the Square"? Such a waste of time...

Why complete the square when we can just use the Quadratic Formula to solve a Quadratic Equation?


Well, one reason is given above, where the new form not only shows us the vertex but makes it easier to solve.

There are also times when the form ax^2 + bx + c may be part of a larger question and rearranging it as a(x+d)^2 + e makes the solution easier, because x only appears once.












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