What is the sum of all values of x that satisfy the equation (x2 -5x+5)x2 +4x-60 =1? Hint: There are three cases to consider
I think I figured out 2 of the cases. The first one being x2 +4x-60=0 while x2 -5x+5=whatever. The second is x2 -5x+5=1 and x2 +4x-60=whatever. But I cant think of a third case
x^4 - 5 x^3 + 5 x^2 + 4 x - 61 = 0
find the roots
try
http://www.wolframalpha.com/input/?i=x^4+-+5+x^3+%2B+5+x^2+%2B+4+x+-+61+%3D+0+&x=5&y=5
To find the sum of all values of x that satisfy the equation, let's consider the three cases mentioned:
Case 1: (x^2 - 5x + 5) = 1 and (x^2 + 4x - 60) = whatever
In this case, we have two separate equations.
To find the values of x in the first equation, we can subtract 1 from both sides:
x^2 - 5x + 5 - 1 = 0
x^2 - 5x + 4 = 0
Next, we can factor the quadratic equation:
(x - 4)(x - 1) = 0
Setting each factor equal to 0 gives us two possible values for x: x = 4 and x = 1.
To find the sum of these two values, we add them together:
x1 + x2 = 4 + 1 = 5
Case 2: (x^2 - 5x + 5) = whatever and (x^2 + 4x - 60) = 1
Similarly, we can solve the first equation by subtracting whatever from both sides:
x^2 - 5x + 5 - whatever = 0
To find the solutions for this equation, we need to know the specific value of "whatever." Without that information, we cannot determine the values of x for this case.
Case 3: (x^2 - 5x + 5) = 1 and (x^2 + 4x - 60) = 0
In this case, we have the equation x^2 + 4x - 60 = 0.
Again, we can factor the quadratic equation:
(x - 6)(x + 10) = 0
Setting each factor equal to 0 gives us two possible values for x: x = 6 and x = -10.
To find the sum of these two values:
x1 + x2 = 6 + (-10) = -4
To find the sum of all values of x that satisfy the original equation, we add the sums of each case:
5 + (-4) = 1
Therefore, the sum of all values of x that satisfy the equation is 1.