how would i factor
(x - 1/x)^2 - 77/12(x - 1/x) + 10 = 0
I assume you mean x - (1/x) and not (x-1)/x, and that 77/12(x - 1/x) means (77/12)[x - (1/x)]. If not, please write your equation in an unambiguous manner, using more parentheses.
Treat (x - 1/x) as a new variable y, and solve the equation
y^2 - (77/12) y + 10 = 0
That cannot be factored easily, so use the quadratic formula to get the two possible value os y. Once you have a number for y, solve the equation
x - 1/x = y to get x
That will require solveing a different quadratic equation. There should be four answers, but no all may be real.
sorry about that. yes i meant x - (1/x).
so i subbed in y for [x - (1/2)]^2, and used the quadratic formula.
i have y = 77/12 +- root of (169/44) all divided by 2.
i'm not sure where to go with this and how i should solve for x. please help
To factor the given equation, we can use a substitution to simplify it. Let's substitute the expression (x - 1/x) with a new variable, let's say y.
Let's rewrite the equation using this substitution:
y^2 - (77/12)y + 10 = 0
Now, we can try to factor this quadratic equation. We need to find two numbers that multiply to give us 10 and add up to give us -77/12.
The factors of 10 are (+/-)1, (+/-)2, (+/-)5, and (+/-)10.
Let's test some combinations to see if any of them add up to -77/12:
1 * 10 = 10 (doesn't work)
2 * 5 = 10 (doesn't work)
-1 * -10 = 10 (doesn't work)
-2 * -5 = 10 (doesn't work)
None of the combinations of factors of 10 add up to give us -77/12. This means the quadratic equation cannot be factored using integers.
However, we can still find the solutions of the equation by using the quadratic formula. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.
In our case, the coefficients are:
a = 1
b = -77/12
c = 10
The quadratic formula states that the solutions of the equation ax^2 + bx + c = 0 can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Let's substitute the values into the formula and solve for x:
x = [(-(-77/12) ± √(((-77/12)^2) - 4(1)(10))) / (2(1)]
Simplifying the equation, we get:
x = [-(77/12) ± √(5929/144 - 40/12)] / 2
x = [-(77/12) ± √(5929/144 - 360/144)] / 2
x = [-(77/12) ± √((5929 - 360)/144)] / 2
x = [-(77/12) ± √(5569/144)] / 2
x = [-(77/12) ± √(5569)/√(144)] / 2
x = [-(77/12) ± √(5569)/12] / 2
x = (-77 ± 12√(5569))/24
Therefore, the solutions to the equation (x - 1/x)^2 - 77/12(x - 1/x) + 10 = 0 are:
x = (-77 + 12√(5569))/24
x = (-77 - 12√(5569))/24