This discussion board is a place to discuss our Daily Challenges and the math and science
related to those challenges. Explanations are more than just a solution — they should
explain the steps and thinking strategies that you used to obtain the solution. Comments
should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .

Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.

Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown

Appears as

*italics* or _italics_

italics

**bold** or __bold__

bold

- bulleted - list

bulleted

list

1. numbered 2. list

numbered

list

Note: you must add a full line of space before and after lists for them to show up correctly

Most simple way to proof: Using Algebra
$\frac {\frac {-b + \sqrt{b^2-4ac}}{2a}}{\frac {-b-\sqrt{b^2-4ac}}{2a}}=\frac{4}{3}\Rightarrow\frac {-b + \sqrt{b^2-4ac}}{-b-\sqrt{b^2-4ac}}=\frac{4}{3}\Rightarrow$ (Cross multiplication) $-3b+3\sqrt{b^2-4ac}=-4b-4\sqrt{b^2-4ac}\Rightarrow b=-7\sqrt{b^2-4ac}\Rightarrow b^2=49(b^2-4ac)\Rightarrow b^2=49b^2-196ac\Rightarrow48b^2=196ac\Rightarrow12b^2=49ac$ [proven in the simplest way]

Let the roots be $4k$ and $3k$. Then we have $4k + 3k = \frac{-b}{a} \implies 7k= \frac{-b}{a}$ and $4k*3k= \frac{c}{a} \implies 12k^2= \frac{c}{a}$. From the first equation $k= \frac{-b}{7a}$. Plugging this value into the second equation, $12* ( \frac{b}{7a})^2 = \frac{c}{a} \implies 12b^2= 49ac$ [proven].

His proof works, just replace the first two statements to $4k+3k=-\frac{b}{a}$ and $4k*3k=\frac{c}{a}$ and do the exact same steps: $k=-\frac{b}{7a}$. Plug in:$12*(-\frac{b}{7a})^2=\frac{c}{a}$, which simplifies to $12b^2=49ac$.

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in`\(`

...`\)`

or`\[`

...`\]`

to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestMost simple way to proof: Using Algebra $\frac {\frac {-b + \sqrt{b^2-4ac}}{2a}}{\frac {-b-\sqrt{b^2-4ac}}{2a}}=\frac{4}{3}\Rightarrow\frac {-b + \sqrt{b^2-4ac}}{-b-\sqrt{b^2-4ac}}=\frac{4}{3}\Rightarrow$ (Cross multiplication) $-3b+3\sqrt{b^2-4ac}=-4b-4\sqrt{b^2-4ac}\Rightarrow b=-7\sqrt{b^2-4ac}\Rightarrow b^2=49(b^2-4ac)\Rightarrow b^2=49b^2-196ac\Rightarrow48b^2=196ac\Rightarrow12b^2=49ac$ [proven

in the simplest way]Log in to reply

Sure seems like a lot of algebra to me. Not that simple. Not that elegant.

Log in to reply

This can be proved by Vieta's formula!!!

Log in to reply

Let the roots be $4k$ and $3k$. Then we have $4k + 3k = \frac{-b}{a} \implies 7k= \frac{-b}{a}$ and $4k*3k= \frac{c}{a} \implies 12k^2= \frac{c}{a}$. From the first equation $k= \frac{-b}{7a}$. Plugging this value into the second equation, $12* ( \frac{b}{7a})^2 = \frac{c}{a} \implies 12b^2= 49ac$ [proven].

Log in to reply

YOU DID NOT PROVE 12b^2=49ac you proved 12b^2=49c sorry i forgot putting the coffecient of x^2

(a)

Log in to reply

His proof works, just replace the first two statements to $4k+3k=-\frac{b}{a}$ and $4k*3k=\frac{c}{a}$ and do the exact same steps: $k=-\frac{b}{7a}$. Plug in:$12*(-\frac{b}{7a})^2=\frac{c}{a}$, which simplifies to $12b^2=49ac$.

Log in to reply

Log in to reply

What about factoring in the a into the problem?

Log in to reply

Neat problem. Nice proof. I'll write one up in a day or two to let others solve the problem.

Log in to reply