A simple proof ?

If $f(x)$ is a quadratic equation with $2$ real roots, prove that the points where the graph of $y=f(x)$ cuts the $x-$ axis are equidistant to every point on the symmetric axis of $f(x)$ .

#Proofs #QuadraticPolynomial

Easy Math Editor

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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

consider the function to be $F(x) = ax^2 + bx + c$ given that roots are real .... two cases arise

(1) * they are equal *

i.e they both lie on axis of symmetry and share the same point hence are equidistant.

(2) they are distinct

the formula for roots is given by $x = \frac{-b}{2a} + or - \frac{\sqrt{b^2-4ac}}{2a}$ ......... {1}

which shows that equal amounts of $\frac{\sqrt{b^2-4ac}}{2a}$ is added and subtracted from the axis of symmetry giving rise to the roots. {this tells that the x coordinates of roots are placed equally apart from axis of symmetry}

they lie on the same horizontal axis {implies y coordinate w.r.t point on axis of symmetry is same} hence roots are equidistant.

Abhinav Raichur - 6 years, 1 month ago

You are really close to the actual proof, you just need to prove that the quadratic has its symmetric axis given by the plot of the equation $x = \dfrac{-b}{2a}$ in the $xy$ plane.

Venkata Karthik Bandaru - 6 years, 1 month ago

oops i considered that to be true ........ we can write F(x) as $F(x) = (x+\frac{b}{2a} )^2 + \frac{b^2 - 4ac}{4 a^2}$ comparing this with the graph of $f(x) = x^2 + k$ which has symmetry at x=0 {the y axis} , after translation of $\frac{b}{2a}$ the symmetry shifts to $x=\frac{-b}{2a}$

Abhinav Raichur - 6 years, 1 month ago

@Abhinav Raichur – Well excellent proof Abhinav Raichur !

Venkata Karthik Bandaru - 6 years, 1 month ago

@Venkata Karthik Bandaru – thank you :)

Abhinav Raichur - 6 years, 1 month 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}`	$\boxed{123}$