Question

if x^3+1÷x^3 = 34√5
prove that x = 2+√5​

Answer 1

Answer:

The easy way to prove this would be to plug in the value for x into the above equation to get the same answer on both sides of the equation.

So, we know that: x=√5+2. All we do is plug this equation for x into x^3+1/x^3 =34√5 for each value of x. It might get a little messy, so I am going to solve this step-by-step.

First, I am going to plug x in: (√5+2)^3+1/(√5+2)^3 =34√5

Now I am going to expand the left side. Because x is being raised to a power of 3 twice, I will just solve for it once, and plug it in:

(√5+2)^3=(√5+2)(√5+2)(√5+2)

=(√5+2)(5+4√5+4)

= (5√5+4*5+4√5+10+8√5+8) = (38+17√5)

Now I can rewrite the equation as: (38+17√5)+1/(38+17√5) =34√5

I’m gonna move the first portion (in front of the addition sign) of the left hand side of the equation to the right hand side:

1/(38+17√5) =34√5-(38+17√5)

1/(38+17√5) =34√5-38-17√5) -> 1/(38+17√5) =17√5-38

If you notice, I should be able to multiply the right hand side by the denominator of the equation on the left hand side and be left with and equation that is equal to 1:

1 =(17√5-38)(38+17√5)

1=646√5+289*5–1444–646√5

1=1

Because we have ended the equation with the same value on each side of the it (they are “one-to-one”), we have successfully proved that √5+2 is in fact the value for x.