Derivation of the Quadratic Formula
This article will just be a quick proof of the quadratic formula , the formula that is used to work out the solutions to the general quadratic equation . The primary method by which this general solution will be derived will be through the method of completing the square.
- Completing the Square
- The Derivation
Completing the Square
The method of completing the square is important to deriving the quadratic formula. We are concerned with deriving the values of x given a quadratic equation in the form:
To “complete the square” for a second-order polynomial, the procedure is as follows:
- Divide the a, b, and c coefficients by the a coefficient.
- Transpose the c coefficient to the other side of the equation.
- Take the b coefficient, and multiply it by one-half (1/2), and let’s store the result in the variable ε. Then, take ε, square it, and add the squared result to both sides of the equation.
- Factor the trinomial expression. The result should be (x + ε)²
- Extract the roots from both sides of the equation
Example with arbitrary second-order polynomial:
We first divide the entire trinomial by the “2” coefficient:
Then, we transpose the “5” onto the other side of the equation:
Next, we multiply -9/2 by 1/2, and store the result (-9/4) as ε. We take ε and square it to get 81/16. We add the result onto both sides of the equation:
Carrying on, we factor the expression, where the solution to the factor will be (x + ε)². We know that ε is equal to (-9/4), so we therefore have:
Finally, we just need to extract the roots of the equation. Take the square root of both equations, which will give us:
Solving for x, we have:
To derive the quadratic formula, we just need to apply the method of completing the square to the quadratic equation’s general form. We are going to manipulate the equation algebraically. Given:
We must first divide the trinomial by the coefficient a, which gives us:
Then, we are going to transpose c/a to the right side of the equation, giving us:
Next, we are going to multiply the b/a coefficient by 1/2, store the result in ε, and square it.
Add ε² to both sides of the equation:
Then, we factor out the trinomial, the result is (x + ε)²
Do the arithmetic on the right-hand side of the equation:
Extract the roots from both sides of the equation:
We finally just move the b/2a coefficient from the left side onto the right side of the equation to get:
“Completing the Square” is a great method to solve for quadratic equations when factoring methods will not suffice. Completing the square can also be used to solve Gaussian integrals , but that’s for another article for the future ;-)
Also, I have a cheat sheet for College Algebra and Trigonometry (that took me ten seconds to make, currently incomplete). Check it out!
- Boljanovic, V. (2016) pp. 89–92
- Wolfram MathWorld’s introduction to Quadratic Equation: http://mathworld.wolfram.com/QuadraticFormula.html
- After Savov, I. (2017), pp. 27–29
- And after my College Algebra and Trigonometry Professor ;-)
- “QM Handout — Gaussian Integrals”: http://www.maths.dur.ac.uk/~dma0wjz/QM/gaussian.pdf
- Boljanovic, V. (2016). Applied Mathematical and Physical Formulas (2nd ed.). South Norwalk: Industrial. (ISBN: 978–0–8311–3592–8)
- Savov, I. (2017). No Bullshit Guide to Linear Algebra. Montréal, Québec: Minireference. (ISBN: 978-0-9920-0102-5)