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6x + 3y – 2x – 5y
Bring the terms containing x together
Bring the terms containing y together
= (6x-2x) + (3y-5y)
= 4x + (-2y)
= 4x - 2y

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Let the consecutive numbers be n and (n+1) respectively.
Their sum is (n + n + 1) = (2n + 1)
Four times their sum = 4(2n + 1)
The product of the numbers = n(n + 1)
The product of two consecutive integers is 4 less than four times their sum.
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So, the equation becomes
n(n + 1) = 4 (2n + 1) - 4
n^2 + n = 8n + 4 - 4
n^2 + n = 8n
Subtract 8n on both the sides
n^2 + n - 8n = 8n - 8n
n^2 - 7n = 0
n (n - 7) = 0
Use the zero product rule
n=0 or (n-7)=0
n=0 or n=7.
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If n=0, n+1 = 0 + 1 = 1.
So, one pair of consecutive integers is 0 and 1.
If n=7, n+1 = 7 + 1 = 8.
So, second pair of consecutive integers is 7 and 8.
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Cheers,
Mastermath.

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(4p^2q)(p^2q^3)
= 4(p^2p^2)(q^3q)
= 4(p^(2+2))(q^(3+1))
Ans:= 4p^4 q^4

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1. f(x) = 2x – 5
So, put x=1,2,4,8,16 and find the value of f(x)
f(1) = 2(1) - 5 = 2 - 5 = -3
f(2) = 2(2) - 5 = 4 - 5 = -1
f(4) = 2(4) - 5 = 8 - 5 = 3
f(8) = 2(8) - 5 = 16 - 5 = 11
f(16) = 2(16) - 5 = 32 - 5 = 27
So, as the value of x increases, f(x) also increases.
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2. f(x) = x2 - 4x + 3
So, put x=1,2,4,8,16 and find the value of f(x)
f(1) = (1)^2 - 4(1) + 3 = 1 - 4 + 3 = 0
f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1
f(4) = (4)^2 - 4(4) + 3 = 16 - 16 + 3 = 3
f(8) = (8)^2 - 4(8) + 3 = 64 - 32 + 3 = 35
f(16) = (16)^2 - 4(16) + 3 = 256 - 64 + 3 = 195
So,initially as the value of x increases, f(x) initially decreases from f(1) to f(2), but after that, as x increases, f(x) increases drastically.
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3. f(x) = x3 + 4x2 - 2x - 3
So, put x=1,2,4,8,16 and find the value of f(x)
f(1) = (1)^3 - 4(1)^2 -2(1) - 3 = 1 - 4(1) -2 - 3 = 1 - 4 - 5 = -8
f(2) = (2)^3 - 4(2)^2 -2(2) - 3 = 8 - 4(4) -4 - 3 = 8 - 16 - 7 = -15
f(4) = (4)^3 - 4(4)^2 -2(4) - 3 = 64 - 4(16) -8 - 3 = 64 - 64 - 11= -11
f(8) = (8)^3 - 4(8)^2 -2(8) - 3 = 512 - 4(64) -16 - 3 = 512 - 256 - 19 = 237
f(16) = (16)^3 - 4(16)^2 -2(16) - 3 = 4096 - 4(256) -32 - 3 = 4096 - 1024 - 35 = 3037
So,initially as the value of x increases, f(x) initially decreases from f(1) to f(2), but after that, as x increases, f(x) increases drastically.
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4. f(x) = 7x
So, put x=1,2,4,8,16 and find the value of f(x)
f(1) = 7(1) = 7
f(2) = 7(2) = 14
f(4) = 7(4) = 28
f(8) = 7(8) = 56
f(16) = 7(16) = 112
So, as the value of x increases, f(x) also increases.
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5. f(x) = log x
So, put x=1,2,4,8,16 and find the value of f(x)
f(1) = log(1) = 0
f(2) = log(2) = 0.301
f(4) = log(4) = 0.602
f(8) = log(8) = 0.903
f(16) = log(16) = 1.204
So, as the value of x increases, f(x) also increases. But the increase is very less in value compared to the above functions.
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f(x)=2x^2-7x+3
(a) f(2)
answer: -3
show your work here:
To find f(2), substitute 2 in place of x
f(2)=2(2)^2-7(2)+3
f(2)=2(4)-14+3
f(2)=8-14+3
f(2)=-3
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(b) f(-3)
answer: 42
show your work here:
To find f(-3), substitute -3 in place of x
f(-3)=2(-3)^2-7(-3)+3
f(-3)=2(9)+21+3
f(-3)=18+21+3
f(-3)=42

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5x^4y-10x^2y^2
We can take 5x^2y common from the two terms
5x^2y(x^2-2y)Ans:

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Let the three consecutive odd integers be:
1st number = x
2nd number = x+2
3rd number = x+4
three time the middle integer is seven more then the sum of the first and third integer.
The sum of the first and the third integer is x+(x+4)= 2x+4.
The three times the middle integer is 3(x+2).
So, the equation becomes
3(x+2) = 7+ (2x+4)
3x + 6 = 7 + 2x + 4
3x + 6 = 11 + 2x
now, subtract 2x on both the sides
3x + 6 - 2x = 11 + 2x - 2x
x + 6 = 11
now, subtract 6 on both the sides
x + 6 - 6 = 11 - 6
x = 5
So, 1st number = x = 5
2nd number = x+2 = 5 + 2 = 7
3rd number = x+4 = 5 + 4 = 9.
Ans: So, the three consecutive odd integers are 5,7 and 9.
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Proof:
Summation of 1st and 3rd number
=5 + 9 = 14
Adding 7 to it we get
14 + 7 = 21.
Three times second number = 7(3) = 21.

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Let x and (x+2) be the two consecutive even numbers.
Their product is x(x+2).
The product of two consecutive odd numbers is 12 more than the square of the smaller number.
The smaller number is x.
so, the equation becomes
x(x+2)= x²+ 12
x² + 2x = x² + 12
We can subtract x² on both the sides.
2x=12
now divide with 2 on both the sides
x=6.
x+2=6+2=8.
So, the smaller odd number is 6.
And the larger odd number is 8.
Proof:
The product of two consecutive even numbers is 12 more than the square of the smaller number.
The product of the numbers is 6 and 8 is 48.
Square of the smaller number (here 6) is 36.
So, add 12 to 36
so, 12 + 36 = 48.

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x²+ 9x = -20
First, add 20 on both the sides
x²+ 9x + 20 = -20 + 20
x²+ 9x + 20 = 0
Now split 9x as sum of 4x and 5x
x²+ 4x + 5x +20=0
Now, take x common from the first two terms.
And take 5 common from the next two terms.
x(x+4)+5(x+4)=0
(x+4)(x+5)=0
Using the zero-product rule
If the product of two numbers is 0, then either of the numbers is 0
so, (x+4)=0 or (x+5)=0
thus, x=-4 or x=-5
Ans: x=-4,-5
Proof:
Put x=-4 in the equation
x²+ 9x= (-4)²+9(-4) = 16 - 36 = -20
now put x=-5 in the equation
x²+ 9x= (-5)²+9(-5) = 25 - 45 = -20
Thus, both the values are satisfied.

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Let x and (x+2) be the two consecutive odd numbers.
Their product is x(x+2).
The product of two consecutive odd numbers is 10 more than the square of the smaller number.
The smaller number is x.
so, the equation becomes
x(x+2)= x²+ 10
x² + 2x = x² + 10
We can subtract x² on both the sides.
2x=10
now divide with 2 on both the sides
x=5.
x+2=5+2=7.
So, the smaller odd number is 5.
And the larger odd number is 7.

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Let the number of cities visited by Rock band be x.
The jazz band visited 96 cities more than the rock band.
So, the number of cities visited by Jazz band will be (x+96).
Together the two tours visited 174 cities.
So, add them together.
Rock + Jazz = 174
x+(x+96)=174
2x+96=174
Now, subtract 96 on both the sides
2x+96-96=174-96
2x=78
Now, divide with 2 on both the sides
x=39
Thus, x+96 = 39+96 =135.
So, the number of cities visited by Rock band is 39.
And the number of cities visited by Jazz band is 135.

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Let the number of cities visited by Rock band be x.
The jazz band visited 96 cities more than the rock band.
So, the number of cities visited by Jazz band will be (x+96).
Together, the two tours visited 174 cities.
So, add both of them
Rock + Jazz = 174
x+(x+96)=174
2x+96=174
subtract 96 on both the sides
2x+96-96=174-96
2x=78
Divide with 2 on both the sides
x=39
so, x+96 = 39+96 = 135
Thus,the number of cities visited by Rock band is 39.
And the number of cities visited by Jazz band is 135.

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2(x+3)-5=x+7
First remove the brackets using distributive property
2x+2(3)-5 = x+7
2x+6-5=x+7
2x+1=x+7
Now subtract x on both the sides
2x+1-x= x+7-x
x+1=7
Subtract 1 on both the sides
x+1-1=7-1
x=6

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(-w²-9w+4-(8w²-4)+(6w²-w+9)
First remove the brackets
-w²-9w+4-8w²+4+6w²-w+9
Now bring the like terms together
The terms containing w and w²
(-w²-8w²+6w²)-9w-w+9
(-9w²+6w²)-10w+9
-3w²-10w+9