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Changing the Form of Expressions

We've already seen one way of changing the form of a quadratic, by factoring. This works especially well when a quadratic factors into a perfect square, as shown below.

x^2+2nx+n^2 = (x+n)^2

If the quadratic isn't a perfect square, we can still rewrite it by completing the square to rewrite it in terms of a perfect square term and another one:

ax^2 + bx + c = a(x-h)^2 + k

To do this, we must find the number n that we saw in the perfect square. Note that the coefficient of x there is two times n. Therefore, we can find n by dividing the coefficient of the x term by 2. Once we know n, we know that the constant term in the perfect square must be n-squared. This is the rule for completing the square, below.

Completing the square
To complete the square in the expression
x^2 + bx + c
divide the coefficient of x by two to get (b/2), then add and subtract this squared, and then factor it:
x^2 + bx + c = x^2 + bx + (b^2/4 - b^2/4) + c = (x^2 + bx + b^2/4) - b^2/4 + c = (x + b/2)^2 - b^2/4 + c
To complete the square for
ax^2 + bx + c
first factor out a and then complete the square.
Examples
Note for each what the term b is, and how we add and subtract to get a perfect square.
  1. Rewrite
    x^2 - 10 x + 4
    in the form
    a(x-h)^2 + k:
    x^2 - 10x + 4 = x^2 - 10x + (25 - 25) + 4 = (x^2 - 10x + 25) - 25 + 4 = (x-5)^2 - 21
  2. Complete the square:
    h(x) = 5x^2 + 30x - 10
    Factor out the 5:
    h(x) = 5(x^2 + 6x - 2)
    Then complete the square:
    x^2 + 6x - 2 = x^2 + 6x + (9 - 9) - 2 = (x^2 + 6x + 9) - 9 - 2 = (x + 3)^2 - 11.
    Thus
    h(x) = 5x^2 + 30x - 10 = 5(x^2 + 6x - 2) = 5((x+3)^2 - 11) = 5(x+3)^2 - 55.

The quadratic formula

The quadratic formula, giving the general solution for the zeros of
r(x) = ax^2 + bx + c,
with a not zero, is found by completing the square. Zeros are where r(x)=0, so we have:

ax^2 + bx + c = 0, or a(x^2 + (b/a)x + (c/a) = 0. a ne 0, so x^2 + (b/a)x + (c/a) = 0.
Then to complete the square we add and subtract ((1/2)(b/a))^2 = (b/(2a))^2 = (b^2/(4a^2)):
x^2 + (b/a)x + (b^2/(4a^2)) - (b^2/(4a^2)) + c/a = 0, or (x^2 + (b/a)x + (b^2/(4a^2))) - (b^2/(4a^2)) + c/a = 0, so (x + b/(2a))^2 - (b^2 - 4ac)/(4a^2) = 0.
(the last term in this last step is because -b^2/(4a^2) + (c/a) = -b^2/(4a^2) + 4ac/(4a^2) = -(b^2 - 4ac)/(4a^2)). Then
(x+b/2a)^2=(b^2-4ac)/(4a^2), so (x+b/2a) = +/- sqrt((b^2-4ac)/(4a^2)) = +/-sqrt(b^2 - 4ac)/(2a), and x = -(b/2a) +/- sqrt(b^2 - 4ac)/(2a) = (-b +/- sqrt(b^2 - 4ac))/(2a).
Practice

Be sure you've gone through each step in the examples above before doing these. Once you've worked them until you're sure that you understand them, go on to the next section. There are more problems of this type in the section test at the end of the section.

Note that you can get new practice problems by clicking the "Refresh" button at the bottom of the practice set.

  1. Rewrite the following in the form
    a(x-h)^2 + k
    a x ^2 - d x + g   =   ( x - ) ^2 +
    (Enter answers, then click: . Answer message: )
  2. Rewrite the following in the same form,
    a(x-h)^2 + k
    a x ^2 + d x + g   =   ( x - ) ^2 +
    (Enter answers, then click: . Answer message: )
For more practice, click: .
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precal: 5.2 - rewriting quadratics
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