Need help with this quadratic math equation...

Question: (x+1/x)^2 - 14(x+1/x) = 72
ended up with:
x^2 + 1 - 14x^3 - 14x -70x^2 = 0

Where do I go from here? I'm trying to find a way to apply the quadratic formula to my results, but I'm having trouble. Is anyone able to help? Thank you in advance.

Answer in the book is:
-2±√3, 9±4√59

First, I disagree with your expansion. I get

It should start with x^4, not x^2
And as you say, that's a daunting equation to solve.

How about letting u = x+ 1/x
Then you have
u^2-14u-72 = 0
(u-18)(u+4) = 0
u = 18,-4

So now just solve
x + 1/x = 18
x^2-18x+1 = 0
I don't get the suggested answer for this one.
and
x + 1/x = -4
x^2+4x+1 = 0
This one agrees with the book.

Given equation: (x+1/x)² - 14(x+1/x) = 72 ...............(1)

Use the substitution
u=x+1/x .............................................................................(1a)
then (1) becomes
u² - 14u -72 = 0 .....................................(2)
Factor (2)
(u-18)(u+4)=0
solve for
u=18 or u=-4 .....................................................................(3)

Backsubstitute solutions (3) into (1a)
u=18
x+1/x=18
multiply by x (x ≠ 0)
x^2-18x+1=0 ......................................................................(4a)
Back substitute
u=-4
x^2+4x+1=0 .......................................................................(4b)

Finally, solve 4a and 4b using the quadratic formula to get appropriate answers.

Note: The book answer you posed has an extraneous "9" tagged on at the end.

Thanks guys, I understand how to solve this equation through this method. The only thing I don't get is why we let "u" be "x" in the first place.

To solve the quadratic equation x^2 + 1 - 14x^3 - 14x - 70x^2 = 0, you first need to rearrange the terms to have the equation in the standard quadratic form, which is ax^2 + bx + c = 0.

In this case, you have -14x^3 - 70x^2 + (1 - 14x) = 0.

Next, you want to simplify the equation by combining like terms. Start by adding the coefficients of x^3 and x^2 together:

-14x^3 - 70x^2 + 1 - 14x = 0

Now, your equation becomes:

-14x^3 - 70x^2 - 14x + 1 = 0

To apply the quadratic formula, you need to identify the values of a, b, and c. In your equation, a = -14, b = -14, and c = 1.

The quadratic formula states that for an equation in the form ax^2 + bx + c = 0:

x = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values from your equation, you get:

x = (-(-14) ± √((-14)^2 - 4(-14)(1))) / (2(-14))

Simplifying further:

x = (14 ± √(196 + 56)) / (-28)

The expression within the square root simplifies to:

x = (14 ± √(252)) / (-28)

Square root of 252 can be simplified as √(4 * 63) = √4 * √63 = 2√63.

Therefore, the equation becomes:

x = (14 ± 2√63) / (-28)

Simplifying further:

x = (7 ± √63) / (-14)

Now, to express the answer in the book, note that √63 can be further simplified. Prime factorizing 63, you get √(3 * 3 * 7) = 3√7.

So the final solution is:

x = (7 ± 3√7) / (-14)

To express it in the form mentioned in the book, you can further simplify the expression as:

x = -2 ± √3, 9 ± 4√59

Therefore, the solution to your quadratic equation is x = -2 ± √3 or x = 9 ± 4√59.