1. We can use the difference of squares identity to factorize this expression:
(5y-6)²-81 = [(5y-6) + 9][(5y-6) - 9]
Now we simplify each factor:
[(5y-6) + 9] = 5y + 3
[(5y-6) - 9] = 5y - 15
Therefore,
(5y-6)²-81 = (5y + 3)(5y - 15)
2. We can use the square of a binomial formula to expand the expression:
1 - (2x-1)² = 1 - (2x)² + 2(2x)(-1) + (-1)²
= 1 - 4x² - 4x + 1
= -4x² - 4x + 2
Now, we can factor out a common factor of -2 from each term:
-2(2x² + 2x - 1)
Therefore,
1 - (2x-1)² = -2(2x² + 2x - 1)
3. Let's begin by simplifying both sides of the equation by combining like terms:
(4.5y+9)-(6.2-3.1y)=7.2y+2.8
4.5y + 9 - 6.2 + 3.1y = 7.2y + 2.8
7.6y + 2.8 = 7.2y + 2.8
Now we can isolate the variable on one side of the equation by subtracting 7.2y from both sides:
7.6y - 7.2y = 2.8 - 2.8
0.4y = 0
Finally, we can solve for y by dividing both sides by 0.4:
y = 0
Therefore, the solution to the equation is y = 0.
4. We can use the difference of squares identity to factorize this expression:
(5c-3d)² - 9d² = [(5c-3d) + 3d][(5c-3d) - 3d]
Now we simplify each factor:
[(5c-3d) + 3d] = 5c
[(5c-3d) - 3d] = 5c - 6d
Therefore,
(5c-3d)² - 9d² = (5c)(5c - 6d)
So, we can write the expression as the product of (5c) and (5c - 6d).