Derivation of the Quadratic Formula

Last week we went over how to solve quadratic equations using the quadratic formula. You can see those posts here: Quadratic Formula 1 2

Recall that the solutions to the quadratic equation ax² + bx + c = 0 are

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

At the end of the second post, I posed the following problem:

Challenge Problem: Solve the general quadratic equation ax² + bx + c = 0 by completing the square, and note that this gives a derivation of the quadratic formula.

I will now provide a detailed solution. You may want to review the following posts before reading the following solution:

Square Root Property Completing the Square Solving Quadratic Equations

Solution:

$ax^2+bx+c=0$

$ax^2+bx=-c$

$x^2+\frac{b}{a} x=-\frac{c}{a}$

$x^2+\frac{b}{a} x+(\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2$

$(x+\frac{b}{2a})^2=-\frac{c}{a}+\frac{b^2}{4a^2}$

$(x+\frac{b}{2a})^2=\frac{b^2}{4a^2}-\frac{c}{a} (\frac{4a}{4a})$

$(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}$

$x+\frac{b}{2a}=\pm \frac{\sqrt{b^2-4ac}}{2a}$

$x=-\frac{b}{2a}\pm \frac{\sqrt{b^2-4ac}}{2a}$

$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

The derivation above was challenging. Try to understand each step. If you have any question on a particular step, feel free to post your question in the comments.

If you liked this article, please share it with your Facebook friends:

Comments

comments