Question

. Solve the following quadratic equation both by factoring and by completing the square:

1 / 4x^2 − x = 3.
Which method do you prefer to solve this equation? Justify your answer using algebraic reasoning.

Answer 1

x = 1 or -3/4

factoring is easier for me

Explanation:

Factoring

We rewrite to ...

4x^2 -x -3 = 0

We want factors of (4)(-3) = -12 that have a sum of -1. They are -4 and 3. Then the factorization is ...

(4x -4)(4x +3)/4 = (x -1)(4x +3)

Solutions are values of x that make the factors zero: x = 1, x = -3/4.

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Completing the square

We start by factoring the leading coefficient from the left side:

4(x^2 -x/4) = 3

Now, we add the square of half the x-coefficient (inside parentheses):

4(x^2 -x/4 +1/64) = 3 + 4/64

4(x -1/8)^2 = 49/16 . . . . . . . . . write as squares

x -1/8 = ±√(49/64) = ±7/8

x = 1/8 ± 7/8

x = -3/4 or +1

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Preference

In this instance I prefer factoring, because it deals with whole numbers up until the end. It does not involve adding and subtracting fractions. On this page, it takes fewer steps.

If you're unfamiliar with math facts, such as the factors of 12, then the purely mechanical "completing the square" method may be preferable. It does not require anything other than finding half of a number and then squaring that.