Question regarding derivation of the quadratic formula To derive

fsrlaihq55

fsrlaihq55

Answered question

2022-03-30

Question regarding derivation of the quadratic formula
To derive the quadratic formula, I want to find (m,n) such that
ax2+bx+c+p=(mx+n)2+c
Expanding the RHS reveals
ax2+bx+c+p=m2x2+2mnx+n2+c.
Then, a=m2,b=2mn(m,n)=(±a,±b2a)
From here, isolating and solving for x would be trivial.Would this general method be correct? I'm confused as to the significance of p since p=n2n=±p here as well.
Perhaps it's been a long day or my algebraic skills have worsened, but I just couldn't seem to work out the quadratic formula with this path of thinking; there would always be some extra term...
I'm aware that dividing the entire original equation by a simplifies things a lot, but that should just be a specific case of this general form.

Answer & Explanation

mistemePietsffi

mistemePietsffi

Beginner2022-03-31Added 8 answers

Your formulation isn't quite right. If you're trying to solve ax2+bx+c=0, you could look for (m,n,p) such that
ax2+bx+c=(mx+n)2p.tag1
Setting the LHS of (1) to zero is then the same as setting the RHS to zero, which yields
x=n±pm.
To obtain m,n,p, you proceed as you've done: Expand the RHS of (1) and match coefficients on powers of x.

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