Find two quadratic equations having the given solutions. (There are many correct answers. Use x for the variable. Enter your answers as a comma-separated list of equations.)
4+5√(5), 4-5√(5)
Using the sum and product properties of roots,
sum of roots = 4+5√(5) + 4-5√(5) = 8
product of roots = (4+5√5)(4-5√5)
= 16 - 25(5) = -109
so the simplest possible quadratic is
x^2 - 8x - 109 = 0
check: https://www.wolframalpha.com/input/?i=solve+x%5E2+-+8x+-+109+%3D+0
Oh, quadratic equations, we meet again! Alright, let me put on my mathematician's wig and get to work. So, you're looking for two quadratic equations with solutions 4+5√(5) and 4-5√(5)? Well, I've got just the equations for you:
Equation 1: (x - (4 + 5√(5))) * (x - (4 - 5√(5))) = 0
Equation 2: (x - (4 + 5√(5))) + (x - (4 - 5√(5))) = 0
These equations should do the trick! Now, go out there and solve them like a quadratic-solving superstar!
To find two quadratic equations with the solutions 4+5√5 and 4-5√5, we can use the fact that if (x-a)(x-b) = 0, then x=a or x=b.
Let's call the unknown quadratic equations as f(x) and g(x).
First, using the solution 4+5√5, we have:
x - (4+5√5) = 0
Simplifying, we get:
x - 4 - 5√5 = 0
Similarly, using the solution 4-5√5, we have:
x - (4-5√5) = 0
Simplifying, we get:
x - 4 + 5√5 = 0
Now, let's multiply these two equations to obtain two quadratic equations:
(x - 4 - 5√5)(x - 4 + 5√5) = 0
Expanding this equation, we get:
(x - 4)^2 - (5√5)^2 = 0
Simplifying further, we have:
x^2 - 8x + 16 - 25(5) = 0
x^2 - 8x + 16 - 125 = 0
x^2 - 8x - 109 = 0
Hence, two quadratic equations with the given solutions are:
f(x) = x^2 - 8x - 109
g(x) = x^2 - 8x - 109
Please note that these are just two examples, and there could be other equations as well.
To find two quadratic equations with the given solutions, we can use the fact that if α and β are the solutions to a quadratic equation of the form ax^2 + bx + c = 0, then the equations can be expressed as (x - α)(x - β) = 0.
Given the solutions α = 4 + 5√5 and β = 4 - 5√5, we can use (x - α)(x - β) = 0 to construct two quadratic equations:
1. (x - (4 + 5√5))(x - (4 - 5√5)) = 0
Expanding the equation:
(x - 4 - 5√5)(x - 4 + 5√5) = 0
(x - 4)^2 - (5√5)^2 = 0
(x - 4)^2 - 25 * 5 = 0
(x - 4)^2 - 125 = 0
2. (x - (4 - 5√5))(x - (4 + 5√5)) = 0
Expanding the equation:
(x - 4 + 5√5)(x - 4 - 5√5) = 0
(x - 4)^2 - (5√5)^2 = 0
(x - 4)^2 - 25 * 5 = 0
(x - 4)^2 - 125 = 0
Therefore, two quadratic equations having the given solutions are:
1. (x - 4)^2 - 125 = 0
2. (x - 4)^2 - 125 = 0
Note that there may be more than one correct answer to this question.