Let the quadratic form f (x1, X2, x3) = X1 ^ 2 + 3x2 + 2x3 ^ 2 + 4x1x2 + 2x1x3 + 2x2x3, (1) write the matrix of the quadratic form; (2) transform the quadratic form into the standard form by the formula method, and obtain the corresponding full rank transformation

Let the quadratic form f (x1, X2, x3) = X1 ^ 2 + 3x2 + 2x3 ^ 2 + 4x1x2 + 2x1x3 + 2x2x3, (1) write the matrix of the quadratic form; (2) transform the quadratic form into the standard form by the formula method, and obtain the corresponding full rank transformation

F (x1, X2, x3) = X1 ^ 2 + 3x2 + 2x3 ^ 2 + 4x1x2 + 2x1x3 + 2x2x3 = (x1 + 2x2 + x3) ^ 2-x2 ^ 2 + X3 ^ 2-2x2x3 = (x1 + 2x2 + x3) ^ 2 + (x3-x2) ^ 2-2x2 ^ 2, so the standard form is Y1 ^ 2 + Y2 ^ 2-2y3 ^ 2

Given x2-4x + 1 = 0, find one of x2 + x2

14
x^2-4x+1=0
=> x^2+1=4x
=> (x^2+1)^2=16x^2 => x^4+1=14x^2 ---------①
x^2+(1/x^2)=(1/x^2)(x^4+1) ----------②
Bring (1) into (2)
Simplification is 14
(another way is to abbreviate x + (1 / x) = 4 x ^ 2 + (1 / x ^ 2) = [x + (1 / x)] ^ 2 - 2)