Suppose a and x satisfy x^2 + (a-1/a)x - 1 = 0. Solve for x in terms of a.
(x - 1/a)(x+a) = 0
(x+a)(x-1/a)=0
x=-a, x=1/a
google golden ratio
To solve the quadratic equation x^2 + (a-1/a)x - 1 = 0, we can use the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In our case, a = 1, b = (a-1/a), and c = -1. Substituting these values into the quadratic formula, we get:
x = (-(a-1/a) ± sqrt((a-1/a)^2 - 4(1)(-1))) / (2(1))
Simplifying the equation gives:
x = (-a + 1/a ± sqrt((a^2 - 2 + 1/a^2) + 4)) / 2
Combining like terms:
x = (-a + 1/a ± sqrt((a^2 + 1/a^2) + 2)) / 2
Now we can simplify further by finding a common denominator:
x = (-a^2 + 1 ± sqrt(a^4 + 1 + 2a^2))/ (2a)
Therefore, the solutions for x in terms of a are:
x = (-a^2 + 1 ± sqrt(a^4 + 1 + 2a^2))/ (2a)