"The Discriminant; Equations in Quadratic Form"
Solve over the complex numbers.
1/e + 1/(square root of e) - 6 = 0
I would let √e = x , then
1/x^2 + 1/x - 6 = 0
multiply by x^2
1 + x - 6x^2 = 0
6x^2 - x - 1 = 0
(3x+1)(2x-1) = 0
x = -1/3 or x = 1/2
√e = -1/3 or √e = 1/2
e = 1/9 or e = 1/4
since we "squared" we have to verify all answers.
if e = 1/9
LS = 1/(1/9) + 1/√(1/9) - 6
= 9 + 3 - 6 = 6
RS = 0, so e = 1/9 does not work
if e = 1/4
LS = 4 + 2 - 6 = 0 = RS
so e = 1/4
BTW, most texts and authors would avoid using e as a variable, since e is generally accepted as the Euler constant
Thank you so much! I've been stuck on this problem for over 2 hours
To solve the equation 1/e + 1/(√e) - 6 = 0, we can treat it as a quadratic equation in the variable √e. Let's go through the steps to solve it.
Step 1: Multiply the equation by e to eliminate the denominator:
e * (1/e + 1/(√e) - 6) = e * 0
This simplifies to:
1 + √e - 6e = 0
Step 2: Rearrange the equation in the quadratic form:
√e - 6e + 1 = 0
Step 3: Now we have a quadratic equation in the form of ax^2 + bx + c = 0, where √e can be considered as x.
a = -6, b = 1, c = 1
Step 4: Calculate the discriminant (D):
D = b^2 - 4ac
D = (1)^2 - 4 * (-6) * 1
D = 1 + 24
D = 25
The discriminant D is positive, which means the quadratic equation has two distinct complex solutions.
Step 5: Solve for √e using the quadratic formula:
√e = (-b ± √D) / (2a)
Substituting the values, we get:
√e = (-1 ± √25) / (2 * -6)
√e = (-1 ± 5) / -12
For the two possible solutions, we have:
1. √e = (5 - 1) / -12 = 4 / -12 = -1/3
2. √e = (-5 - 1) / -12 = -6 / -12 = 1/2
Step 6: Square both sides to get e:
1. (√e)^2 = (-1/3)^2
e = 1/9
2. (√e)^2 = (1/2)^2
e = 1/4
So, the solutions to the equation 1/e + 1/(√e) - 6 = 0 over the complex numbers are e = 1/9 and e = 1/4.