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Derive the quadratic formula. In other words, in the equation

$ax^2 + bx + c = 0$

change the subject to $x$ .

#Algebra #QuadraticEquations #Sharky

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Quadratic formula can be derived more easily , we will use tschirnhaus transformation for the quadratic (used for reducing quartics and cubics)

The equation can be written as $x^2+\frac{b}{a}x+\frac{c}{a}=0$ Use the transformation $x=y-\frac{b}{2a}$ $\left(y-\frac{b}{2a}\right)^2+\frac{b}{a}\left(y-\frac{b}{2a}\right)+\frac{c}{a}=0$ $y^2-\frac{b}{a}y+\frac{b^2}{4a^2}+\frac{b}{a}y-\frac{b^2}{2a^2}+\frac{c}{a}=0$ $y^2=\frac{b^2-4ac}{4a^2}$ $x+\frac{b}{2a}=\frac{\pm\sqrt{b^2-4ac}}{2a}$ $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Shriram Lokhande - 6 years, 10 months ago

This uses the interesting property (for those not too familiar with it)

$\displaystyle \begin{cases}x=y-\frac{a_{n-1}}{n\cdot a_n} \\\\ f(x)=a_n x^n+a_{n-1}x^{n-1}+\cdots+a_1 x+a_0,\: a_n\neq 0\end{cases}$

$\displaystyle \implies f(x)=a_n y^n+p_{n-2}y^{n-2}+p_{n-3}y^{n-3}+\cdots +p_1 y+p_0$

mathh mathh - 6 years, 10 months ago

Rewrite: $x^2+2x \cdot\dfrac{b}{2a}=-\dfrac{c}{a}.$ Add $(b/2a)^2$ to both sides and complete square: $x^2+2x \cdot\dfrac{b}{2a}+\left(\dfrac{b}{2a}\right)^2=\left(\dfrac{b}{2a}\right)^2-\dfrac{c}{a} ~\implies \left(x+\dfrac{b}{2a}\right)^2=\dfrac{b^2}{4a^2}-\dfrac{c}{a}=\dfrac{b^2-4ac}{4a^2}.$ Take square root and rearrange: $x+\dfrac{b}{2a}=\dfrac{\pm\sqrt{b^2-4ac}}{2a}~\implies x=\dfrac{-b\pm\sqrt{b^2+4ac}}{2a}.$

Jubayer Nirjhor - 6 years, 10 months ago

congratulation for 135 / 135

sara sharma - 6 years, 10 months ago

I don't know the score yet. Hopefully, it is 135/135.

Sharky Kesa - 6 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$