Quadratic formula: Difference between revisions

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.

Statement

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

Proof

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.

Variation

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: @@
-===General Solution For A Quadratic by Completing the Square===
+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.
-Let the quadratic be in the form <math>a\cdot x^2+b\cdot x+c=0</math>.
+== Statement ==
-Moving c to the other side, we obtain
+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>
-<math>ax^2+bx=-c</math>
+=== Proof ===
+We start with
-Dividing by <math>{a}</math> and adding <math>\frac{b^2}{4a^2}</math> to both sides yields
+<cmath>ax^{2}+bx+c=0</cmath>
-<math>x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}</math>.
+Dividing by <math>a</math>, we get
-Factoring the LHS gives
+<cmath>x^{2}+\frac{b}{a}x+\frac{c}{a}=0</cmath>
-<math>\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}</math>
+Add <math>\frac{b^{2}}{4a^{2}}</math> to both sides in order to complete the square:
-As described above, an equation in this form can be solved, yielding
+<cmath>\left(x^{2}+\frac{b}{a}x+\frac{b^{2}}{4a^{2}}\right)+\frac{c}{a}=\frac{b^{2}}{4a^{2}}</cmath>
-<math>{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}</math>
+Complete the square:
-This formula is also called the [[Quadratic Formula]].
+<cmath>\left(x+\frac{b}{2a}\right)^{2}+\frac{c}{a}=\frac{b^{2}}{4a^{2}}</cmath>
-We simply plug in a, b, and c and out pops the 2 values of x.
+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.
+=== Variation ===
+In some situations, it is preferable to use this variation of the quadratic formula:
+<cmath>\frac{2c}{-b\mp\sqrt{b^2-4ac}}</cmath>
+== See Also ==
+* [[Quadratic equation]]
+[[Category:Algebra]]
+[[Category:Quadratic equations]]
+{{stub}}

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Quadratic formula: Difference between revisions

Latest revision as of 09:11, 2 February 2025

Contents

Statement

Proof

Variation

See Also