Were doing factoring and i don't get these (^ means squared)
1) m^-3m-2
2) 6c^-5c-4
3) ab+mn+an+mb
4) The length of a rectangular garden is 5ft more than the width. The garden has a 3ft wide cement walk sorrounding it. The total area of the garden and the walk is 546ft^. Find the dimensions of the garden
5) Find 2 consecutive integers such that the sum of their squares is 52
6) The perimeter of a rectangle is 22ft and the area is 24ft^. Find the dimensions of the rectangle
4)
GARDEN
width = w
length = w +5
GARDEN w/CEMENT WALK
width = w + 6
length = w + 5 + 6 = w + 11
a = 546 ft sq.
a = lw
a = (w+6)(w+11)
546 = (w+6)(w+11)
546 = w^2 + 17w + 66
0 = w^2 + 17w - 480
w = (-17 +/- sqrt(289 - 4(1)(-480))/2
w = (-17 +/- sqrt(289 + 1920))/2
w = (-17 +/- sqrt(2209))/2
w = (-17 +/- 47)/2
has to be + because it must be positive
w = 30/2 = 15
l = w + 5 = 20
6)
P = 22
A = 24
2l + 2w = 22
lw = 24
l = 24/w
2(24/w) + 2w = 22
48/w + 2w = 22
48 + 2w^2 = 22w
w^2 - 11w + 24 = 0
(w - 8)(w - 3) = 0
w = 8 or 3
l = 3 or 8
dimensions = 3 x 8
3) ab+mn+an+mb
ab + an + mn + mb
a(b + n) + m(n+b)
(a + m)(b + n)
5) No solution. Obviously 4 and 5 give a sum of squares too low (41) and 5 and 6 give a sum of squares that is too high (61).
If you had said two consecutive EVEN integers, the answer would be 4 and 6.
1) To factor the expression m^-3m-2, we can first identify any common factors. In this case, there are no common factors other than 1.
Next, let's rearrange the terms: m^-3m-2 = -2m - m^-3.
To further factor this expression, we can factor out a common factor of m from both terms:
-2m - m^-3 = m(-2 - m^-4).
Therefore, the factored form of the expression m^-3m-2 is m(-2 - m^-4).
2) Similarly, for the expression 6c^-5c-4, let's rearrange the terms: 6c^-5c-4 = 6 - 5c - 4c.
There are no common factors other than 1 here, so we proceed to group like terms:
6 - 5c - 4c = 6 - (5c + 4c).
Next, we combine the like terms inside the parentheses:
6 - (5c + 4c) = 6 - 9c.
Therefore, the factored form of the expression 6c^-5c-4 is 6 - 9c.
3) The expression ab+mn+an+mb does not seem to be factorable because there are no common factors among the terms. Therefore, the factored form remains ab+mn+an+mb.
4) Let's denote the width of the rectangular garden as x. Since the length of the garden is 5ft more than the width, we can express the length as x + 5.
The area of a rectangle is calculated by multiplying the length and width.
Therefore, the area of the garden can be expressed as x(x + 5).
Since there is a 3ft wide cement walk surrounding the garden, we need to add an additional 6ft (3ft on each side) to both the length and the width.
The new dimensions of the garden with the walk are (x + 6) and (x + 5 + 6), which simplifies to x + 6 and x + 11.
The total area of the garden and the walk is given as 546ft^2, so we can set up an equation:
(x + 6)(x + 11) = 546.
Expanding the equation, we have x^2 + 17x + 66 = 546.
Rearranging the equation, we get x^2 + 17x - 480 = 0.
Now we can solve this quadratic equation to find the value of x, which represents the width of the garden.
Once we find x, we can substitute it back into the expressions for the length (x + 5) and the width (x) to determine the dimensions of the garden.
5) Let's call the two consecutive integers x and x+1. The sum of their squares is 52, so we can set up an equation:
x^2 + (x + 1)^2 = 52.
Expanding the equation, we have x^2 + (x^2 + 2x + 1) = 52.
Combining like terms, we get 2x^2 + 2x + 1 = 52.
Rearranging the equation, we have 2x^2 + 2x - 51 = 0.
Now we can solve this quadratic equation to find the values of x, which represent the consecutive integers.
6) Let's denote the length of the rectangle as l and the width as w. The perimeter of a rectangle is calculated as 2(l + w), so we can set up an equation:
2(l + w) = 22.
Simplifying the equation, we have l + w = 11.
The area of a rectangle is calculated by multiplying the length and width, so the area can be expressed as lw.
We're given that the area is 24ft^2, so we have lw = 24.
Now we have a system of simultaneous equations:
l + w = 11 and lw = 24.
We can solve this system of equations to find the values of l and w, which represent the dimensions of the rectangle.