Calculus Examples

$4 - x^{2} = x + 2$

Step 1

Subtract $x$ from both sides of the equation.

$4 - x^{2} - x = 2$

Step 2

Subtract $2$ from both sides of the equation.

$4 - x^{2} - x - 2 = 0$

Step 3

Subtract $2$ from $4$ .

$- x^{2} - x + 2 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $- 1$ out of $- x^{2} - x + 2$ .

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Step 4.1.1

Factor $- 1$ out of $- x^{2}$ .

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

Step 4.1.2

Factor $- 1$ out of $- x$ .

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

Step 4.1.3

Rewrite $2$ as $- 1 (- 2)$ .

$- (x^{2}) - (x) - 1 \cdot - 2 = 0$

Step 4.1.4

Factor $- 1$ out of $- (x^{2}) - (x)$ .

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

Step 4.1.5

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 2)$ .

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

Step 4.2

Factor.

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Step 4.2.1

Factor $x^{2} + x - 2$ using the AC method.

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Step 4.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 4.2.1.2

Write the factored form using these integers.

$- ((x - 1) (x + 2)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$- (x - 1) (x + 2) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$x + 2 = 0$

Step 6

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 7

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 8

The final solution is all the values that make $- (x - 1) (x + 2) = 0$ true.

$x = 1, - 2$