Deriving the Quadratic Formula

Question

In deriving the Quadratic Formula, near the end all the derivations I see convert sqrt(x2) to x. But sqrt(x2) = |x|.

Answer 1

the definition of y=|x|

is y=x,,,x=y,,, for x≥0

and y=-x ,,,,x=-y,,,for x<0

find when ax²+bc+c=0

same as

(x-b/2a)²=b²/4a²-c/a

taking square root both sides, and abiding to absolute value equivalent

|x-b/2a|=√(b²/4a²-c/a),,,when x-b/2a≥0,,,,x-b/2a=+√(b²/4a²-c/a)

when x-b/2a<0,,,,x-b/2a=-√(b²/4a²-c/a)

everything works fine

Answer 2

y = ax^2 +bx + c = 0

complete the square and you get the quadratic formula

a(x^2 +bx/a) = -c

a(x^2 +bx/a + b^2/4a^2) = -c +b^2/4a

a(x+b/2a)^2 = (b^2 -4ac)/4a

divide by a

(x+b/2a)^2 = (b^2-4a)/4a^2

take square roots of both sides

x+b/2a = sqr(b^2-4ac)/2a

x = -b/2a +/- (1/2a)sqr(b^2 -4ac)

x = (-b +/-sqr(b^2 -4ac)/2a

Answer 3

Answer 4

Let us say we have

x² = a

where a ≥ 0. To calculate x, we can take square root of both sides, which as you correctly stated, yields

|x| = √‾a mid. step

Now that we know the absolute value of x, what is the value of x itself?

x = ±√‾a

which means that x = √‾a or x = -√‾a.

In the solutions that you see, they may skip the middle step! That's all.