Proof of the Quadratic Formula
Students often ask me how do Mathematicians come up with various formulas in Trigonometry, Algebra and Calculus? By method of Proofs and Derivation I would exclaim, which in my opinion is becoming a lost art. I found deriving formulas helped to seal the understanding of the concepts and theories in Math. I believe it is thus fitting as an end to our talk on polynomials, to derive the equation of the quadratic formula. We will do so by using the last method we discussed, "Completing the Square".
Remember the standard form for the quadratic equation reads:
f(x) = ax^2 + bx + c
To find the roots of (or solution to) the polynomial we will let f(x) = 0.
We are going to apply the method of completing the square.
STEP 1: Ensure that the coefficient of the squared term in x is 1. Currently it is "a". We do so by dividing the entire polynomial by "a"(both sides of the equation please).
(ax^2)/a + (b/a)x + c/a = 0/a
x^2 +(b/a)x + c/a = 0
STEP 2: Add square of half the coefficient of the second term (which is b/a) to both sides of equation(to keep things balanced). That is add the square of half of b/a.
x^2 + (b/a)*x + (b/2a)^2 +c/a = 0 + (b/2a)^2
x^2 + (b/a)*x + (b/2a)^2 = (b/2a)^2 - c/a
STEP 3: The left side can be unFOIL to a perfect square. Expand the right side of the equation as well.
(x + b/2a)^2 = (b^2/4a^2) - c/a
(x + b/2a)^2 = (b^2 - 4ac)/4a^2
STEP 4: Take the square roots of both sides and solve for x.
x + b/2a = SQRT((b^2 - 4ac)/4a^2)
x = -b/2a + SQRT(b^2 - 4ac)/2a or -b/2a - SQRT(b^2 - 4ac)/2a
x = [-b + SQRT(b^2 - 4ac)]/2a or [-b - SQRT(b^2 - 4ac)]/2a
The quadratic formula has two possible solutions (two possible roots). Why? The maximum number of roots a polynomial (including quadratic equations) is determined by the degree of the polynomial, that is the size of the exponent of the leading term. For example f(x)=x^3-8 has a maximum of three real roots and f(t)=t^2-3t-4 has at most two real roots. Remember a polynomial can have many, one, or no real roots.
We have completed the polynomial series for Algebra II.
Next time we will move on to the more fascinating Exponential and Logarithmic Functions. Many processes around us can be modeled by them. Stay tuned.