factorise
(5+9x-2x^2)/ ((x^2-6x+23)^ (5/2))
To factorize the given expression (5+9x-2x^2)/ ((x^2-6x+23)^ (5/2)), we can follow these steps:
Step 1: Factorize the denominator:
The denominator (x^2-6x+23) can be factored using the quadratic formula or by completing the square. However, in this case, the expression inside the square root is not factorable using real numbers. Therefore, we leave it as it is and move forward.
Step 2: Factorize the numerator:
The numerator (5+9x-2x^2) can be factorized by finding the common factors or using the quadratic formula. In this case, we can use the quadratic formula to solve for the roots of the quadratic equation.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the given quadratic equation, a = -2, b = 9, and c = 5.
Using the quadratic formula, we can find the roots as follows:
x = (-9 ± √(9^2 - 4(-2)(5))) / (2(-2))
x = (-9 ± √(81 + 40)) / (-4)
x = (-9 ± √121) / (-4)
x = (-9 ± 11) / (-4)
So, we have two roots: x = (-9 + 11) / (-4) = 2 / (-4) = -1/2, and x = (-9 - 11) / (-4) = -20 / (-4) = 5.
Therefore, the numerator (5+9x-2x^2) can be factorized as -2(x + 1/2)(x - 5).
Step 3: Simplify the expression:
Now, we can simplify the expression by canceling out common factors between the numerator and the denominator. In this case, we have -2(x + 1/2)(x - 5) in the numerator and (x^2-6x+23)^(5/2) in the denominator.
So, the final factorized expression is:
-2(x + 1/2)(x - 5) / (x^2-6x+23)^(5/2)
That's the factorized form of the given expression.