2. The area of a rectangle is 3 times the area of a square of side (x+1)cm. The length and breadth of the rectangle are (2x+5)cm and (2x-1)cm respectively.(a) Form an equation and shows that it reduces to x2 + 2x - 8 = 0.
(b) Solve the equation in (a)
3.4x2 + 4xy + y2 - z2 over y2+ 2yz + z2 - 4x2 = ?
1. Given that a2 + 7a = b2 +7b, and that a is not equal to b, find the value of 1/14(a+b).
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A (sq) = `l^2`
A(rect) = l x b
`therefore A=(x+1)^2` (square) and
`A=(2x+5) times (2x-1)` (retangle)
But the rectangle is also 3 times the area of the square
`therefore A = 3(x+1)^2` (rectangle)
`therefore 3(x+1)^2 = (2x+5)(2x-1)` (these both represent the area of the rectangle). Now simplify:
`therefore 3(x^2 +2x+1 )= 4x^2-2x+10x -5`
`3x^2 +6x +3 = 4x^2 +8x - 5`
Bring everything to the one side:
`therefore 0 = 4x^2-3x^2+8x -6x -5-3`
`0= x^2+2x-8` which can also be written:
`x^2 +2x -8=0`
Now factorize. Use the factors of `1x^2` and the factors of -8 That is `1x times 1x` and for the 8 1x8 would not work but 4x2 will work. Care to arrange the symbols correctly. You can check your work if you are ever unsure :
Each factor is equal to zero
`therefore x-2=0 and x+4=0`
`therefore` x=2 and x=-4
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