What is the value of x?
(1) 4x² – 4x + 1 = 0
(2) 4x² – 1 = 0


A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(A) The both statements are quadratic equations. A quadratic equation can have one solution, two distinct solutions, or no solution. Let’s solve each one.
4x² – 4x + 1 = 0
(2x)² – 2 × (2x) + 1 = 0
Now we can apply the well-known formula and get the square of the difference.
(2x – 1)² = 0
2x – 1 = 0
x = 1/2
We have the only one value as a solution. Therefore statement (1) by itself is sufficient.

4x² – 1 = 0
(2x)² = 1
2x = 1 or 2x = -1
x = 1/2 or x = -1/2
We have two distinct values as solutions. Therefore statement (2) by itself is NOT sufficient.

Statement (1) by itself is sufficient, while statement (2) by itself is not. The correct answer is A.
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4x² – 1 = 0 can be solved by itself as there is not written anywhere that it must not be a fraction.

So 4x² – 1 = 0
4x² = 1
x² = 1/4
So x = 1/2

Hence the answer should be D.

x² = 1/4 yields two solutions: x = 1/2 and x = -1/2 .

The well-known formula is (a – b)² = a² –2ab + b²

We have (2x)² – 2 × (2x) + 1 = 0
Comparing to the formula, take a = 2x and b = 1:
(2x)² – 2 × (2x) + 1 = 0
a² –2ab + b²

(2x – 1)² = 0
(a – b)² = 0

1. Why?
Our goal is to determine what set of values Statement (1) defines. Statement (1) is a quadratic equation and thus we need to solve it. You may use any method that you like. We solved the equation by factoring.

2. How?
We used the formula (a – b)² = a² –2ab + b² in reverse.
It comes directly from multiplying the two factors in parentheses:
(a – b)(a – b) = aa – ba – ab + (-b)(-b) = a² –2ab + b²

In case of the equation we have:
(2x – 1)² = (2x)² – 2 × 2x + 1

We did it to apply the formula, to see what a is and what b is.

You can square root the both sides, because both x² and 0.25 are not negative. However you do not know if x is negative or positive. So
√x² = |x| , NOT just x.

Thus, we get
√x² = √0.25
|x| = 0.5
x = 0.5 or x = -0.5


By making the step from √x² = √0.25 to x = 0.5 you imposed additional condition that x is positive.

Plug in both 0.5 and -0.5 into the original equation to see that each one fits.