book says:

(4-4.00x + x^2)(6.00x10^-6C)= x^2(15.0x10^-6C)

they got:
3.00x^2+ 8.00x-8= 0

I don't get how they got that quadradic equation. (I mulitplied over and then divided the 15.0x10^-6 over and I didn't get what the book got)

(my problem that I'm doing is similar problem where

(d^2-dx+x^2)q1= x^2(3q)
would I be able to solve for x?
(I don't think so but since all they give is 3q, q, and d for a rod like the one below with a charged particle in equillibrium between the 2 charged outer particles. (one being 3q and other is q)

(3q)o==========o=====o(q)

total distance is given only as d.


Thank you

first part reposted below

(4-4.00x + x^2)(6.00x10^-6C)= x^2(15.0x10^-6C)

3.00x^2+ 8.00x-8= 0 => how did they get this from the equation above?

I tried everything but I must be doing something incorrect.

Thank you.

Divide both sides of the first equation by 6*10^-6C. Here's what I get:

(x^2 - 4x +4) = 2.5 x^2
Now double both sides to get rid of the fractional coefficient 2.5.
2x^2 - 8x +8 = 5x^2
3x^2 +8x -8 = 0
Use the quadratic equation to solve for x.
x = [-8 +/- sqrt 160]/6
= +0.7749 or -3.4415

divide both sides by 2x^(-6C) to get

(4 - 4x + x^2)(2) = 5x^2
8 - 8x + 2x^2 = 5x^2
rearrange:
3x^2 + 8x - 8 = 0

Thank you both drwls and Reiny

I missed where they multiplied by 2 to get rid of the fraction.

To understand how the quadratic equation was obtained in the given problem, let's go through the steps:

Original equation:
(4-4.00x + x^2)(6.00x10^-6C) = x^2(15.0x10^-6C)

First, distribute the multiplication on the left side:
(4 * 6.00x10^-6C - 4.00x * 6.00x10^-6C + x^2 * 6.00x10^-6C) = x^2(15.0x10^-6C)

Now, simplify the terms on the left side:
24.00x10^-6C - 24.00x10^-6C + 6.00x^3 * 10^-12C = 15.0x^2 * 10^-6C

The 24.00x10^-6C terms cancel each other out, leaving:
6.00x^3 * 10^-12C = 15.0x^2 * 10^-6C

Divide both sides by 10^-12C to remove the units:
6.00x^3 = 15.0x^2 * 10^6

Now, divide both sides by x^2 to isolate the term with x:
6.00x = 15.0 * 10^6

Divide both sides by 6.00 to solve for x:
x = (15.0 * 10^6) / 6.00

Simplify the right side:
x ≈ 2.5 * 10^6

So, the value of x that satisfies the initial equation is approximately 2.5 * 10^6.

Now, let's examine your problem:

(d^2 - dx + x^2)q1 = x^2(3q)

In this case, you have a similar quadratic equation. You have a constant term, a linear term, and a quadratic term containing x. However, in this equation, there is no common factor that can be divided out to simplify it into a quadratic equation easily.

Therefore, you won't be able to directly solve for x with the given equation. It seems like you may need additional information or conditions to solve for x in this specific problem.