Express x^2 − 4x + 5 in the form (x + a)^2 + b and hence, or otherwise, write down the coordinates of the minimum point on the curve.
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`x^2 - 4x + 5`
To express in (x + a)^2 + b form we need to complete the square.
`(x^2 - 4x +c)^2 + 5` To find c, divide -4 by 2 and square it.
`(x^2 - 4x +4) + 5-4`
`(x - 2)^2 + 1`
Hence in the form (a+x)^2 + b we have: `(x-2)^2 + 1`
This makes (2, 1) the minimum point.
We know that;
`(x-2)^2 = x^2-4x+4`
`(x-2)^2+1 = x^2-4x+4+1`
`(x-2)^2+1 = x^2-4x+5`
We know that `(x-2)>= 0` always .So the minimum of y will be obtained when `(x-2) =0`
`(x-2) = 0`
So the required answers are;
`x^2-4x+5 = (x-2)^2+1`
minimum point on curve is 1.
`x^2 - 4x + 5 `
`(x^2 - 4x) + 5`
use the formula (-b)/2 and the result squared will be c
`(-4)/2 = -2^2= 4`
`(x^2 -4x+4) + 5-4`
`(x^2-4x+4) + 1`
use the square root of a and c (use the sign between a and b), the number on the outside is the y point.
the minimum point would be 2,1 or just 1
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