Help: Absolute Value

Consider all real numbers $c$ such that $|x-8|+|4-x²|=c$ has exactly $3$ real solutions. Find the sum of all possible values of $c$ .

#Algebra

Easy Math Editor

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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

Hope this helps

Jason Gomez - 3 months, 1 week ago

And at y<6 there are no solutions to the equation

Jason Gomez - 3 months, 1 week ago

Thanks a lot mate, this is a huge help. I finally understand it. :)

Ryan Merino - 3 months, 1 week ago

Np bro, i had used my modulus arithmetic after a long time thanks to you

Jason Gomez - 3 months, 1 week ago

This visual graph might help you(it’s possible and quite easy to graph this with pen and paper), the lines y=10 and y=12.25 denote the only c for which there are only three solutions, the y=10 is easy to figure out on pen and paper while the y=12.25 will require you to take a derivative

Jason Gomez - 3 months, 3 weeks ago

Sorry if I'm very slow mate but I still haven't get it even after many tries.

Ryan Merino - 3 months, 2 weeks ago

And forgot to mention y and c are the same thing

Jason Gomez - 3 months, 1 week 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}`	$\boxed{123}$