Simplify x / 6x - x^2
Please show me how you did it so i can understand!
Another one, Simplify x^2 - 3x - 18 / x + 3
This is a fraction
Another one, k + 3 / 4k - 2 (this is a fraction) * 12k^2 + 2k - 4)
x/(6x-x^2)
= x / x(6-x)
= 1/(6-x)
(x^2 - 3x - 18) / (x + 3)
= (x-5)(x+3)/(x+3)
= x-5
(k+3)/(4k-2) * (12k^2+2k-4)
= (k+3)/(2(2k-1)) * 2(2k-1)(3k+2)
= (k+3)/(3k+2)
for 1 and 3, they are right, but for 2, the answer choices are
x-3,
x-6 where x doesn't equal -3
and x-6 where x doesn't equal 6
for #2, Steve meant to say:
(x-6)(x+3)/(x+3)
= x-6 , x ≠ -3
Yes, I neglected to mention excluded values. One cannot gloss over stuff like that. My bad.
To simplify the expressions, let's break them down step by step.
1. Simplify x / (6x - x^2):
To simplify this expression, we can factor out an 'x' from both the numerator and denominator:
x / [x(6 - x)]
Now we can cancel out the 'x' terms:
1 / (6 - x)
Therefore, x / (6x - x^2) simplifies to 1 / (6 - x).
2. Simplify (x^2 - 3x - 18) / (x + 3):
The numerator is a quadratic expression that isn't easily factored, so let's use the quadratic formula to find the roots of the numerator:
x = [-b ± √(b^2 - 4ac)] / (2a)
In this case, a = 1, b = -3, and c = -18. Plugging in these values gives us:
x = [3 ± √((-3)^2 - 4(1)(-18))] / (2*1)
x = [3 ± √(9 + 72)] / 2
x = [3 ± √81] / 2
x = [3 ± 9] / 2
Thus, the roots are x = (3 + 9) / 2 = 6 and x = (3 - 9) / 2 = -3.
Now we can factor the numerator using these roots:
(x - 6)(x + 3) / (x + 3)
We can cancel out the (x + 3) terms:
(x - 6) / 1
x - 6
Therefore, (x^2 - 3x - 18) / (x + 3) simplifies to x - 6.
3. Simplify (k + 3) / (4k - 2) * (12k^2 + 2k - 4):
To simplify this expression, we can first factor the second part of the expression, (12k^2 + 2k - 4).
Factor out a 2 from each term:
2(6k^2 + k - 2)
Now let's factor the quadratic expression inside the parentheses using the quadratic formula:
k = [-b ± √(b^2 - 4ac)] / (2a)
In this case, a = 6, b = 1, and c = -2. Plugging in these values gives us:
k = [-1 ± √(1^2 - 4(6)(-2))] / (2*6)
k = [-1 ± √(1 + 48)] / 12
k = [-1 ± √49] / 12
k = [-1 ± 7] / 12
The roots are k = (-1 + 7) / 12 = 6/12 = 1/2 and k = (-1 - 7) / 12 = -8/12 = -2/3.
Now we can factor the quadratic expression:
2(k - 1/2)(k + 2/3)
Now let's simplify the overall expression:
(k + 3) / (4k - 2) * (2(k - 1/2)(k + 2/3))
We can cancel out the (k - 1/2) terms:
(k + 3) / (4k - 2) * (2(k + 2/3))
Now simplify the expression:
(k + 3) * (k + 2/3)
To multiply, distribute the k and the 2/3 to each term inside the parentheses:
k^2 + (2/3)k + (3)(k) + (3)(2/3)
Combine like terms:
k^2 + (2/3)k + 3k + 2
Simplify the expression:
k^2 + (11/3)k + 2
Therefore, (k + 3) / (4k - 2) * (12k^2 + 2k - 4) simplifies to k^2 + (11/3)k + 2.