Difference between revisions of "Quadratic formula"

Latest revision as of 09:11, 2 February 2025

The quadratic formula is a general expression for the solutions to a quadratic equation. It is used when other methods, such as completing the square, factoring, and square root property do not work or are too tedious.

For any quadratic equation $ax^2+bx+c=0$ , the following equation holds. $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

We start with

$ax^{2}+bx+c=0$

Dividing by $a$ , we get

$x^{2}+\frac{b}{a}x+\frac{c}{a}=0$

Add $\frac{b^{2}}{4a^{2}}$ to both sides in order to complete the square:

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

Complete the square:

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

Move $\frac{c}{a}$ to the other side:

$\left(x+\frac{b}{2a}\right)^{2}=\frac{b^{2}}{4a^{2}}-\frac{c}{a}=\frac{ab^{2}-4a^{2}c}{4a^{3}}=\frac{b^{2}-4ac}{4a^{2}}$

Take the square root of both sides:

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

Finally, move the $\frac{b}{2a}$ to the other side:

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

This is the quadratic formula, and we are done.

In some situations, it is preferable to use this variation of the quadratic formula:

$\frac{2c}{-b\mp\sqrt{b^2-4ac}}$

@@ Line 1: / Line 1: @@
 The '''quadratic formula''' is a general [[expression]] for the [[root (polynomial)|solutions]] to a [[quadratic equation]]. It is used when other methods, such as [[completing the square]], [[factoring]], and [[square root property]] do not work or are too tedious.
-===General Solution For A Quadratic by Completing the Square===
+== Statement ==
+For any quadratic equation <math>ax^2+bx+c=0</math>, the following equation holds.
+<cmath>x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}</cmath>
+=== Proof ===
 We start with
 <cmath>ax^{2}+bx+c=0</cmath>
-Divide by <math>a</math>:
+Dividing by <math>a</math>, we get
 <cmath>x^{2}+\frac{b}{a}x+\frac{c}{a}=0</cmath>
@@ Line 31: / Line 36: @@
 This is the quadratic formula, and we are done.
-===Video Solution===
-https://youtu.be/Akz8LcVGj5k
 === Variation ===
 In some situations, it is preferable to use this variation of the quadratic formula:
-<cmath>\frac{2c}{-b\pm\sqrt{b^2-4ac}}</cmath>
+<cmath>\frac{2c}{-b\mp\sqrt{b^2-4ac}}</cmath>
 == See Also ==
@@ Line 45: / Line 47: @@
 [[Category:Algebra]]
 [[Category:Quadratic equations]]
+{{stub}}