Can you use square roots to solve all quadratic equations?

1 Answer
Dec 22, 2014

If the question is about using the square root directly against the equation, the answer is definitely NO.

However, with certain transformation of a given equation into a different but equivalent form it is possible. Here is the idea.

Assume, for example, the same equation as analyzed in the previous answer:
x^2+x=63

If we could transform it to something like y^2=b then the square root of both sides would deliver a solution.
So, let's transform our equation to this form.
Expression x^2+x is not a square of anything, but x^2+x+1/4 is a square of x+1/2 because
(x+1/2)^2=x^2+2*x*1/2+1/4=x^2+x+1/4

Therefore, it is reasonable to transform the original equation into
(x+1/2)^2-1/4=63 or
(x+1/2)^2=253/4
From the last equation, which is absolutely equivalent to the original one, using the operation of the square root, we derive two linear equations:
x+1/2=sqrt(253)/2 and x+1/2=-sqrt(253)/2

So, two solutions are:
x=(-1+sqrt(253))/2 and x=(-1-sqrt(253))/2

The above method is pretty universal and handy if you don't remember a formula for solutions of a quadratic equation. Let me illustrate this with another example.
-3x^2+2x+8=0

Step 1. Divide everything by -3 to have x^2 with a multiplier 1:
x^2-2/3x-8/3=0

Step 2. Since a coefficient at x is -2/3, use (x-1/3)^2 in a transformed equation:
(x-1/3)^2-1/9-8/3=0 or
(x-1/3)^2=25/9

Step 3. Use square root:
x-1/3=5/3 and x-1/3=-5/3

Step 4. Solutions:
x=6/3=2 and x=-4/3