Calculus Examples

$x + {(y - 1)}^{2} = 0$

Step 1

Subtract ${(y - 1)}^{2}$ from both sides of the equation.

$x = - {(y - 1)}^{2}$

Step 2

Simplify both sides.

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Step 2.1

Rewrite ${(y - 1)}^{2}$ as $(y - 1) (y - 1)$ .

$x = - ((y - 1) (y - 1))$

Step 2.2

Expand $(y - 1) (y - 1)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$x = - (y (y - 1) - 1 (y - 1))$

Step 2.2.2

Apply the distributive property.

$x = - (y \cdot y + y \cdot - 1 - 1 (y - 1))$

Step 2.2.3

Apply the distributive property.

$x = - (y \cdot y + y \cdot - 1 - 1 y - 1 \cdot - 1)$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $y$ by $y$ .

$x = - (y^{2} + y \cdot - 1 - 1 y - 1 \cdot - 1)$

Step 2.3.1.2

Move $- 1$ to the left of $y$ .

$x = - (y^{2} - 1 \cdot y - 1 y - 1 \cdot - 1)$

Step 2.3.1.3

Rewrite $- 1 y$ as $- y$ .

$x = - (y^{2} - y - 1 y - 1 \cdot - 1)$

Step 2.3.1.4

Rewrite $- 1 y$ as $- y$ .

$x = - (y^{2} - y - y - 1 \cdot - 1)$

Step 2.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.3.2

Subtract $y$ from $- y$ .

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

Step 2.4

Apply the distributive property.

$x = - y^{2} - (- 2 y) - 1 \cdot 1$

Step 2.5

Simplify.

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Step 2.5.1

Multiply $- 2$ by $- 1$ .

$x = - y^{2} + 2 y - 1 \cdot 1$

Step 2.5.2

Multiply $- 1$ by $1$ .

$x = - y^{2} + 2 y - 1$

Step 3

To rewrite as a function of $y$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $y$ is on the other side.

$f (y) = - {(y - 1)}^{2}$