X^4-13X^3+31X^2+126 find the roots

solving x^4 - 13x^3 + 31x^2 + 126 = 0 is not easy.

Wolfram is a powerful website that gives us the following:
http://www.wolframalpha.com/input/?i=solve+x%5E4+-+13x%5E3+%2B+31x%5E2+%2B+126%3D0

Notice we have 2 reals, and 2 complex roots.

Since we are given exact real roots of 7 ± √7, suggests that Wolfram used a formula
There is indeed such a formula but it is extremely complicated. Google "solving quartic equations"

If you know Calculus, you should try Netwon's Method

To find the roots of the equation X^4-13X^3+31X^2+126, we can use the factoring method or the numerical method like the quadratic formula.

1. Factoring method:
First, let's group the terms together:
X^4-13X^3+31X^2+126 = (X^4-13X^3) + (31X^2+126)

We notice that we can factor out an X^3 from the first grouping:
X^4-13X^3 = X^3(X-13)

And similarly, we can factor out a 31 from the second grouping:
31X^2+126 = 31(X^2+4)

Now our equation becomes:
(X^3)(X-13) + 31(X^2+4) = 0

Now, we have two factors, (X^3)(X-13) and 31(X^2+4), which are equal to zero. So, we set each factor equal to zero and solve for X individually:

Setting X^3 = 0:
X^3 = 0
X = 0

Setting X-13 = 0:
X - 13 = 0
X = 13

Setting X^2 + 4 = 0:
X^2 + 4 = 0
X^2 = -4

The last equation has no real solutions because the square of a real number cannot be negative.

Therefore, the roots of the equation X^4-13X^3+31X^2+126 are X = 0 and X = 13.

2. Numerical method using the quadratic formula:
If factoring is not possible, we can use the quadratic formula for finding the roots of X^4-13X^3+31X^2+126.

The general form of the quadratic equation is AX^2 + BX + C = 0, where A, B, and C are coefficients.

In our equation, the coefficients are:
A = 1
B = -13
C = 31

Now, we can substitute these values into the quadratic formula:
X = (-B ± √(B^2 - 4AC)) / 2A

Plugging in the values:
X = (13 ± √((-13)^2 - 4 * 1 * 31)) / (2 * 1)
X = (13 ± √(169 - 124)) / 2
X = (13 ± √45) / 2

Simplifying further:
X ≈ (13 + √45) / 2 ≈ 11.388
X ≈ (13 - √45) / 2 ≈ 1.612

Thus, the roots of the equation X^4-13X^3+31X^2+126 are approximately X ≈ 11.388 and X ≈ 1.612.