Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Quadratic formula"

m (not much point in having a page link to itself)
(ccc)
 
(19 intermediate revisions by 12 users not shown)
Line 1: Line 1:
−
The '''quadratic formula''' is a general expression for the solutions to a [[quadratic equation]].
+
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===
 +
We start with
  
−
===General Solution For A Quadratic by Completing the Square===
+
<cmath>ax^{2}+bx+c=0</cmath>
 +
 
 +
Divide by <math>a</math>:
 +
 
 +
<cmath>x^{2}+\frac{b}{a}x+\frac{c}{a}=0</cmath>
 +
 
 +
Add <math>\frac{b^{2}}{4a^{2}}</math> to both sides in order to complete the square:
 +
 
 +
<cmath>\left(x^{2}+\frac{b}{a}x+\frac{b^{2}}{4a^{2}}\right)+\frac{c}{a}=\frac{b^{2}}{4a^{2}}</cmath>
 +
 
 +
Complete the square:
 +
 
 +
<cmath>\left(x+\frac{b}{2a}\right)^{2}+\frac{c}{a}=\frac{b^{2}}{4a^{2}}</cmath>
 +
 
 +
Move <math>\frac{c}{a}</math> to the other side:
  
−
Let the quadratic be in the form <math>a\cdot x^2+b\cdot x+c=0</math>.
+
<cmath>\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}}</cmath>
  
−
Moving c to the other side, we obtain
+
Take the square root of both sides:
  
−
<math>a\cdot x^2+b\cdot x=-c</math>
+
<cmath>x+\frac{b}{2a}=\pm\sqrt{\frac{b^{2}-4ac}{4a^{2}}}=\frac{\pm\sqrt{b^{2}-4ac}}{2a}</cmath>
  
−
Dividing by <math>{a}</math> and adding <math>\frac{b^2}{4a^2}</math> to both sides yields
+
Finally, move the <math>\frac{b}{2a}</math> to the other side:
  
−
<math>x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}</math>.
+
<cmath>x=-\frac{b}{2a}+\frac{\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}</cmath>
  
−
Factoring the LHS gives
+
This is the quadratic formula, and we are done.
  
−
<math>\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}</math>
+
===Video Solution by ligonmathkid2===
 +
https://youtu.be/Akz8LcVGj5k
  
−
As described above, an equation in this form can be solved, yielding
+
=== Variation ===
 +
In some situations, it is preferable to use this variation of the quadratic formula:
  
−
<math>{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}</math>
+
<cmath>\frac{2c}{-b\pm\sqrt{b^2-4ac}}</cmath>
  
−
This formula is also called the quadratic formula.
+
== See Also ==
 +
* [[Quadratic formula]]
 +
* [[Quadratic equation]]
  
−
Given the values <math>{a},{b},{c}</math>, we can find all [[real]] and [[complex number|complex]] solutions to the quadratic equation.
+
[[Category:Algebra]]
 +
[[Category:Quadratic equations]]
 +
[https://artofproblemsolving.com/wiki/index.php/TOTO_SLOT_:_SITUS_TOTO_SLOT_MAXWIN_TERBAIK_DAN_TERPERCAYA TOTO SLOT]

Latest revision as of 15:55, 19 February 2024

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.

General Solution For A Quadratic by Completing the Square

We start with

\[ax^{2}+bx+c=0\]

Divide by $a$:

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

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.

Video Solution by ligonmathkid2

https://youtu.be/Akz8LcVGj5k

Variation

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

\[\frac{2c}{-b\pm\sqrt{b^2-4ac}}\]

See Also

TOTO SLOT

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Quadratic_formula&oldid=215819"