Difference between revisions of "Quadratic formula"

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(General Solution For A Quadratic by Completing the Square)
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The '''quadratic formula''' is a general expression for the solutions to a [[quadratic equation]].
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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.
  
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===General Solution For A Quadratic by Completing the Square===
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Let the quadratic be in the form <math>ax^2+bx+c=0</math>. Moving <math>c</math> to the other side, we obtain
  
===General Solution For A Quadratic by Completing the Square===
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<cmath>ax^2+bx=-c.</cmath>
  
Let the quadratic be in the form <math>a\cdot x^2+b\cdot x+c=0</math>.
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Dividing by <math>{a}</math> and adding <math>\frac{b^2}{4a^2}</math> to both sides yields
  
Moving c to the other side, we obtain
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<cmath>x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}.</cmath>
  
<math>a\cdot x^2+b\cdot x=-c</math>
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Completing the square on the left-hand side gives
  
Dividing by <math>{a}</math> and adding <math>\frac{b^2}{4a^2}</math> to both sides yields
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<cmath>\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}.</cmath>
  
<math>x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}</math>.
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As described above, an equation in this form can be solved, yielding
  
Factoring the LHS gives
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<cmath>{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}.</cmath>
  
<math>\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}</math>
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This formula is also called the quadratic formula. Given the values <math>{a},{b},{c},</math> we can find all [[real]] and [[complex number|complex]] solutions to the quadratic equation.
  
As described above, an equation in this form can be solved, yielding
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=== Variation ===
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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>
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<cmath>\frac{2c}{-b\pm\sqrt{b^2-4ac}}</cmath>
  
This formula is also called the [[Quadratic Formula]].
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== See Also ==
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* [[Quadratic formula]]
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* [[Quadratic equation]]
  
Given the values <math>{a},{b},{c}</math>, we can find all [[real]] and [[complex number|complex]] solutions to the quadratic equation.
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[[Category:Algebra]]
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[[Category:Quadratic equations]]

Revision as of 19:28, 12 December 2021

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

Let the quadratic be in the form $ax^2+bx+c=0$. Moving $c$ to the other side, we obtain

\[ax^2+bx=-c.\]

Dividing by ${a}$ and adding $\frac{b^2}{4a^2}$ to both sides yields

\[x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}.\]

Completing the square on the left-hand side gives

\[\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}.\]

As described above, an equation in this form can be solved, yielding

\[{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}.\]

This formula is also called the quadratic formula. Given the values ${a},{b},{c},$ we can find all real and complex solutions to the quadratic equation.

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