# Quadratic formula by completing the square - easier method

By Murray Bourne, 14 Feb 2011

Most math text books derive the Quadratic Formula as follows:

To find the roots of a quadratic equation in the form

ax^{2}+bx+c= 0, follow these steps:(i) If

adoes not equal 1, divide each side bya(so that the coefficient of thex^{2}is 1).

(ii) Rewrite the equation with the

constantterm on the right side.

(iii) Complete the square by adding the square of one-half of the coefficient of

x(this is the square ofb/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 [1]):

Multiply both sides by 4*a*:

Now, go back to the starting equation [1], find the coefficient of *x* (it's *b*) and square it (we get *b*^{2}). 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 4*a* = 12. We multiply both sides by 12:

Add 48 to both sides:

Now, in the question, *b* = −2. We square this (*b*^{2} = 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.]

See the 14 Comments below.