simplify and state your restrictions for the following expression (show all your work):
a) x^2-x-20/x^2-6x divided by x^2+9x+20/x^2-12x+36
b) 2x^2+3x+1/4x^2-9 minus x^2-3x-1/9-4x^2
a) To simplify the expression, we need to factor the numerators and denominators and cancel out any common factors.
First, let's factor the numerators and denominators separately:
Numerator of the first fraction: x^2 - x - 20
= (x - 5)(x + 4)
Denominator of the first fraction: x^2 - 6x
= x(x - 6)
Numerator of the second fraction: x^2 + 9x + 20
= (x + 4)(x + 5)
Denominator of the second fraction: x^2 - 12x + 36
= (x - 6)(x - 6)
= (x - 6)^2
Now we can rewrite the expression as:
[(x - 5)(x + 4) / x(x - 6)] ÷ [(x + 4)(x + 5) / (x - 6)^2]
To divide fractions, we invert the second fraction (the divisor) and multiply:
[(x - 5)(x + 4) / x(x - 6)] * [(x - 6)^2 / (x + 4)(x + 5)]
Next, we can cancel out common factors:
[(x - 5)(x + 4) / x(x - 6)] * [(x - 6) / (x + 5)]
Finally, we simplify further by canceling out (x - 6) from the numerator and denominator:
(x - 5)(x + 4) / x(x + 5)
So, the simplified expression is (x - 5)(x + 4) / x(x + 5).
The restriction is that x cannot be equal to 0 or -5 because these values would result in division by zero.
b) To simplify the expression, we need to combine like terms.
The numerator of the first fraction: 2x^2 + 3x + 1
The denominator of the first fraction: 4x^2 - 9
The numerator of the second fraction: x^2 - 3x - 1
The denominator of the second fraction: 9 - 4x^2
Now we can rewrite the expression as:
(2x^2 + 3x + 1) / (4x^2 - 9) - (x^2 - 3x - 1) / (9 - 4x^2)
To subtract fractions, we need a common denominator.
The common denominator for the two fractions is (4x^2 - 9)(9 - 4x^2).
Now we can rewrite the expression with a common denominator:
[(2x^2 + 3x + 1)(9 - 4x^2) - (x^2 - 3x - 1)(4x^2 - 9)] / [(4x^2 - 9)(9 - 4x^2)]
Next, we expand and simplify the numerator:
[(18x^2 + 27x + 9 - 8x^4 - 12x^3 - 4x^2 - 4x^2 + 12x + 3) - (4x^4 - 12x^3 - 4x^2 + 9x^2 - 27x - 9)] / [(4x^2 - 9)(9 - 4x^2)]
Combine like terms in the numerator:
[18x^2 + 27x + 9 - 8x^4 - 12x^3 - 4x^2 - 4x^2 + 12x + 3 - 4x^4 + 12x^3 + 4x^2 - 9x^2 + 27x + 9] / [(4x^2 - 9)(9 - 4x^2)]
Combine like terms further:
[14x^2 - x^4 + 9] / [(4x^2 - 9)(9 - 4x^2)]
So, the simplified expression is (14x^2 - x^4 + 9) / [(4x^2 - 9)(9 - 4x^2)].
The restrictions are that x cannot be equal to 9/2 or -9/4 because these values would result in division by zero. Also, x cannot be equal to ±3/2 because these values would result in the denominator (4x^2 - 9) being equal to zero.