Deriving the Quadratic Formula

When we enter algebra, one of the most important formulas we learn is the quadratic formula, which is used to find the zeros of functions. We're often told to memorize it, but why does this equation actually work? Continue reading to see the breakdown of the well-known quadratic formula!

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It is established that the standard form of a quadratic equation is:

Firstly, we will divide the whole equation by a, getting:

Next, we will bring over the constant term (c/a) to complete the square with the remaining x terms:

To complete the square, divide the x term's (b/a) coefficient by 2 to get (b/2a), then square it, which will give you the constant term. Remember to add to both sides, like so:

Then simplify:

Then, combine the terms on the right side of the equation. To combine fractions, you multiply the terms as needed with fractions equivalent to 1/1 that will give them the same denominator. In this case, (-c/a) can be multiplied by (4a/4a) to get the denominator of 4(a^2), which is the denominator of the second term:

Next, simplify again by adding the two terms:

The numerator of the fraction on the right side of the equation looks recognizable! Now, you take the square root of both sides to isolate the x (as a reminder, the zeros of the equation are represented by the variable x):

Simplify once more:

Bring over the (b/2a) term:

Then, simplify one last time by combining the terms with the same denominator:

Which finally gives you the quadratic formula! :)

And that is why this formula can be used to find the zeros of any quadratic equation!

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Written by: Janice Le