Find the points of intersection for the graph of x^2-2x-y=6 and x-y=-4 but without using a calculator

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To find the points of intersection for the graph of `x^2-2x-y=6` and `x-y=-4` we have to solve the two equations.

`x^2-2x-y=6`

`rArr y=x^2-2x-6` ..........(i)

Again, `x-y=-4`

`rArr y=x+4` .............(ii)

Substitute `(x+4)` for `y` in equation (i) to get:

`x+4=x^2-2x-6`

`rArr x^2-3x-10=0`

`rArr x^2-5x+2x-10=0`

`rArr x(x-5)+2(x-5)=0`

`rArr (x-5)(x+2)=0`

Hence, `x=5,-2`

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check:for equation (i),  `f(5)=5^2-2*5-6=9`

                                    `f(-2)=(-2)^2-2*(-2)-6=2`

for equation (ii),  `f(5)=5+4=9`

                           `f(-2)=-2+4=2`

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So when x = 5, y = 9 for both equations, and when x = -2, y = 2 for both equations.

Therefore, the two points of intersection are (5,9) and (-2,2).

The graph:

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