Differenciate with respect to x. Y = (x-1) √x² + 2x +2

1 Simplify  \sqrt{{x}^{2}}

x

2

​

  to  xx.

y=(x-1)x+2x+2

y=(x−1)x+2x+2

2 Regroup terms.

y=x(x-1)+2x+2

y=x(x−1)+2x+2

3 Expand.

y={x}^{2}-x+2x+2

y=x

2

−x+2x+2

4 Simplify  {x}^{2}-x+2x+2x

2

−x+2x+2  to  {x}^{2}+x+2x

2

+x+2.

y={x}^{2}+x+2

y=x

2

+x+2

5 Move all terms to one side.

y-{x}^{2}-x-2=0

y−x

2

−x−2=0

6 Use the Quadratic Formula.

x=\frac{1+\sqrt{4y-7}}{-2},\frac{1-\sqrt{4y-7}}{-2}

x=

−2

1+

4y−7

​

​

,

−2

1−

4y−7

​

​

7 Simplify solutions.

x=-\frac{1+\sqrt{4y-7}}{2},-\frac{1-\sqrt{4y-7}}{2}

x=−

2

1+

4y−7

​

​

,−

2

1−

4y−7

​

​

Done