Calculus Examples

$3 - x^{2} = x - 17$

Step 1

Subtract $x$ from both sides of the equation.

$3 - x^{2} - x = - 17$

Step 2

Add $17$ to both sides of the equation.

$3 - x^{2} - x + 17 = 0$

Step 3

Add $3$ and $17$ .

$- x^{2} - x + 20 = 0$

Step 4

Factor the left side of the equation.

Tap for more steps...

Step 4.1

Factor $- 1$ out of $- x^{2} - x + 20$ .

Tap for more steps...

Step 4.1.1

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

$- (x^{2}) - x + 20 = 0$

Step 4.1.2

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

$- (x^{2}) - (x) + 20 = 0$

Step 4.1.3

Rewrite $20$ as $- 1 (- 20)$ .

$- (x^{2}) - (x) - 1 \cdot - 20 = 0$

Step 4.1.4

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

$- (x^{2} + x) - 1 \cdot - 20 = 0$

Step 4.1.5

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 20)$ .

$- (x^{2} + x - 20) = 0$

Step 4.2

Factor.

Tap for more steps...

Step 4.2.1

Factor $x^{2} + x - 20$ using the AC method.

Tap for more steps...

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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 4.2.1.2

Write the factored form using these integers.

$- ((x - 4) (x + 5)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$- (x - 4) (x + 5) = 0$

Step 5

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

$x - 4 = 0$

$x + 5 = 0$

Step 6

Set $x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 7

Set $x + 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 7.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 8

The final solution is all the values that make $- (x - 4) (x + 5) = 0$ true.

$x = 4, - 5$