complete the square

complete the square

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It is required to complete the square root

x^{2 }-2x

= (x^{2 }-2x+1)-1 (adding and subracting 1)

= (x-1)^{2 }- 1 because ( x-y)^{2 }= x^{2 }+y^{2
}-2xy

and here we get the result

I hope it helps !!

Hello Shelby,

Recall your algebraic identities

(x + y)^{2} = x^{2} + 2xy + y^{2}

(x - y)^{2} = x^{2} - 2xy + y^{2}

We will use these identities to do Completing the square. We have x^{2} - 2x

To make it a perfect square we will do 2y = -2, so we get y = -1, add square of y to make it a perfect square. y^{2} = 1. So, we have x^{2} - 2x + 1 = (x - 1)^{2} (answer)

To generalize this to any quadratic function of the form x^{2} + bx. In this, we have 2y = b which gives

y = b/2. Therefore,

x^{2} + bx + (b/2)^{2} = (x + (b/2))^{2}

Quadratic function of the form x^{2} - bx. In this, we have 2y = -b which gives y = -b/2. Therefore,

x^{2} - bx + (-b/2)^{2} = (x - (-b/2))^{2}

I hope this helps.