Quadratic formula by completing the square – easier method
[14 Feb 2011]
Most math text books derive the Quadratic Formula as follows:
To find the roots of a quadratic equation in the form ax2 + bx + c = 0, follow these steps:
(i) If a does not equal 1, divide each side by a (so that the coefficient of the x2 is 1).
(ii) Rewrite the equation with the constant term on the right side.
(iii) Complete the square by adding the square of one-half of the coefficient of x (this is the square of b/2a) to both sides.
(iv) Write the left side as a perfect square and simplify the right side.
(v) Equate and solve.
Now, that’s pretty messy since there is a lot going on.
Alternative Derivation of Quadratic Formula
Here’s a simpler process.
Once again, we start with an equation in the form (and call it Equation ):
Multiply both sides by 4a:
Now, go back to the starting equation , find the coefficient of x (it’s b) and square it (we get b2). Add that number to both sides of our equation:
Write the left side as a perfect square:
Solve for x:
I hope you find that easier to follow than the more common method (presented at top).
Solving a quadratic equation using the alternative method of completing the square
Question: Solve the quadratic equation using completing the square:
Answer: In this example. a = 3, so 4a = 12. We multiply both sides by 12:
Add 48 to both sides:
Now, in the question, b = −2. We square this (b2 = 4) and add it to both sides:
Next, write the left side as a perfect square:
Solve for x:
What are your thoughts on this method? Is it easier for you?
[Hat tip to reader Lemmie, who sent me this method.]