Difference between revisions of "Quadratic formula"
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− | The '''quadratic formula''' is a general expression for the [[ | + | 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 | ||
− | === | + | <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: | ||
− | + | <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> | |
− | + | Take the square root of both sides: | |
− | < | + | <cmath>x+\frac{b}{2a}=\pm\sqrt{\frac{b^{2}-4ac}{4a^{2}}}=\frac{\pm\sqrt{b^{2}-4ac}}{2a}</cmath> |
− | + | Finally, move the <math>\frac{b}{2a}</math> to the other side: | |
− | < | + | <cmath>x=-\frac{b}{2a}+\frac{\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}</cmath> |
− | + | 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: | ||
− | < | + | <cmath>\frac{2c}{-b\pm\sqrt{b^2-4ac}}</cmath> |
− | + | == See Also == | |
+ | * [[Quadratic formula]] | ||
+ | * [[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.
Contents
General Solution For A Quadratic by Completing the Square
We start with
Divide by :
Add to both sides in order to complete the square:
Complete the square:
Move to the other side:
Take the square root of both sides:
Finally, move the to the other side:
This is the quadratic formula, and we are done.
Video Solution by ligonmathkid2
Variation
In some situations, it is preferable to use this variation of the quadratic formula: