Since it has to be the least value I've chosen the solution with "–".
The question does NOT ask to find the least value of x. It asks to find such value of x that f(x) will be the least.
I have solved for x and I got the A solution.
A is a trap choice. Apparently you solved 0 = x² + 2bx + 4, which corresponds to f(x) = 0. But, why must the least value of f(x) be 0? There are no logical grounds for that. The least value of f(x) can also be positive or negative depending on the value of b. Pick some values for b between -2 and 2. Then pick some values for b greater than 2. You'll see that the least value of f(x), which is (4 – b²), will have a different sign.
Furthermore, choice A can NOT be applied if b is between -2 and 2, while question statement asks "for any b". That is a hint that choice A is NOT a correct one.
Here is the image of general view of such parabolas that a quadratic function can be. Take a look at their bottom points that correspond to the least value for each parabola. Such point can be below the x-axis (negative), above the x-axis (positive) or lie on the x-axis (zero).
(4 – b²) is not a square (it is equal to (2 – b)(2 + b))
Indeed. Furthermore, it is negative if |b| > 2 . But how does it affect the reasoning?
x² + 2bx + 4 = (x² + 2bx + b²) + 4 – b² Here I have added and deducted b², leaving the whole function the same (when we add 0, which is b² – b², it does not change anything).
Now let's denote 4 – b² as A. It will be easier to pay attention to the expression within parentheses.
(x² + 2bx + b²) + A = (x + b)² + A
I have applied the regular formula (a + c)² = a² + 2ac + c² .
So now we have f(x) = (x + b)² + A
A is some constant, it doesn't change when x changes. Thus the value of the function is the least when (x + b)² is the least. It is a square, so the least value it can possibly be is 0. Let's see if it can be 0 and what the value of x will be in that case. (x + b)² = 0 x = -b
What is the "clue" that tells we should add and subtract b²?
x² + 2bx is the clue.
We see that it has: x² – the square of the variable, which tells us it is a square function 2 × x × b – the term with the variable (not squared), which should correspond to 2 × a × c in the formula for the square of a sum (a + c)² = a² + 2ac + c² in our case we want to transform the equation into having this formula (x + ??)² = x² + 2 × x × ?? + ??²
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