Question

hi! this looks like a VERY basic algebra question but it something different! pwease help ಠ ∩ ಠ
Teachers told us to learn the value of x but i'm curious and wanna know how to find it..
(this calculation is actually related to finding \(p^H\) of dilute acids )

Answer 1

\(10^{-14}=(x+10^{-7})x\)

find \(x\) without using any calculator

Answer 2

i think that quadratic formula can't help us with this one..

Answer 3

This is a quadratic equation in x.
You can use the quadratic formula.
Start by multiplying out the right side using the distributive property.

Answer 4

but that becomes very lengthy tho

Answer 5

Not at all.
It's just a quadratic with 3 terms.

Answer 6

Why does that matter? If it's a solution, then go find it. (It is NOT lengthy.)

Answer 7

Copy the left side.
Then distribute x on the right side.
Can you do it and show what you got?

Answer 8

First off, just look at it. x = 0 is ALMOST a solution.

Answer 9

yeah i'm typing

Answer 10

\(10^{-14}=(x+10^{-7})x\)

\(x^2+10^{-7}x-10^{-14}=0\)

\(x=\Large \frac{-10^{-7} \pm\sqrt{1\color{red}{-4\times (-10^{-14})}}}{2}\)

Answer 11

should i ignore the red part? it is almost 0

(we need approximate solutions but not 0)

Answer 12

Not quite. Check that "1" under the radical.

Answer 13

In the root, there should be b^2.
You have 1.

Answer 14

oops

*\(x=\Large \frac{-10^{-7} \pm\sqrt{10^{-14}\color{red}{-4\times (-10^{-14})}}}{2}\)

Answer 15

\(\Large x= \frac{-10^{-7} \pm\sqrt{(10^{-7})^2 -4\times (-10^{-14})}}{2}\)

Answer 16

Yes, good.
Now simplify the radical.

Answer 17

\(x=\Large \frac{-10^{-7} \pm\sqrt{10^{-14}\color{red}{+4\times (10^{-14})}}}{2}\)

\(x=\Large \frac{-10^{-7} \pm\sqrt{5\times (1{0^{-14})}}}{2}\)

\(x=\Large \frac{-10^{-7} \pm10^{-7}\sqrt{5}}{2}\)

Answer 18

\(\Large x= \frac{-10^{-7} \pm\sqrt{5 \times 10^{-14}}}{2}\)

\(\Large x= \frac{-10^{-7} \pm\sqrt{5} \times 10^{-7}}{2}\)

Good.

Answer 19

\(x=\Large \frac{-10^{-7} + 10^{-7}\sqrt{5}}{2}\) (x is positive ->given)
\(x=\Large \frac{10^{-7}( \sqrt{5}-1) }{2}\)

\(x=0.618 \times 10^{-7}\)

Answer 20

thanks! :) this is the value of x I was looking for!

i think its better to learn the value because it is used very often in these pH calculations

Answer 21

\(x = 6.18 \times 10^{-8} \)

Answer 22

Great.
You're welcome.