Precalculus Examples

$| x - 10 | = x^{2} - 10 x$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 10 x = | x - 10 |$

Step 2

Rewrite the equation as $| x - 10 | = x^{2} - 10 x$ .

$| x - 10 | = x^{2} - 10 x$

Step 3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 10 = \pm (x^{2} - 10 x)$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 10 = x^{2} - 10 x$

Step 4.2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 10 x = x - 10$

Step 4.3

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 4.3.1

Subtract $x$ from both sides of the equation.

$x^{2} - 10 x - x = - 10$

Step 4.3.2

Subtract $x$ from $- 10 x$ .

$x^{2} - 11 x = - 10$

Step 4.4

Add $10$ to both sides of the equation.

$x^{2} - 11 x + 10 = 0$

Step 4.5

Factor $x^{2} - 11 x + 10$ using the AC method.

Tap for more steps...

Step 4.5.1

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 $10$ and whose sum is $- 11$ .

$- 10, - 1$

Step 4.5.2

Write the factored form using these integers.

$(x - 10) (x - 1) = 0$

Step 4.6

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

$x - 10 = 0$

$x - 1 = 0$

Step 4.7

Set $x - 10$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.7.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 4.7.2

Add $10$ to both sides of the equation.

$x = 10$

Step 4.8

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

Tap for more steps...

Step 4.8.1

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

$x - 1 = 0$

Step 4.8.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.9

The final solution is all the values that make $(x - 10) (x - 1) = 0$ true.

$x = 10, 1$

Step 4.10

Next, use the negative value of the $\pm$ to find the second solution.

$x - 10 = - (x^{2} - 10 x)$

Step 4.11

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- (x^{2} - 10 x) = x - 10$

Step 4.12

Simplify $- (x^{2} - 10 x)$ .

Tap for more steps...

Step 4.12.1

Rewrite.

$0 + 0 - (x^{2} - 10 x) = x - 10$

Step 4.12.2

Simplify by adding zeros.

$- (x^{2} - 10 x) = x - 10$

Step 4.12.3

Apply the distributive property.

$- x^{2} - (- 10 x) = x - 10$

Step 4.12.4

Multiply $- 10$ by $- 1$ .

$- x^{2} + 10 x = x - 10$

Step 4.13

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 4.13.1

Subtract $x$ from both sides of the equation.

$- x^{2} + 10 x - x = - 10$

Step 4.13.2

Subtract $x$ from $10 x$ .

$- x^{2} + 9 x = - 10$

Step 4.14

Add $10$ to both sides of the equation.

$- x^{2} + 9 x + 10 = 0$

Step 4.15

Factor the left side of the equation.

Tap for more steps...

Step 4.15.1

Factor $- 1$ out of $- x^{2} + 9 x + 10$ .

Tap for more steps...

Step 4.15.1.1

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

$- (x^{2}) + 9 x + 10 = 0$

Step 4.15.1.2

Factor $- 1$ out of $9 x$ .

$- (x^{2}) - (- 9 x) + 10 = 0$

Step 4.15.1.3

Rewrite $10$ as $- 1 (- 10)$ .

$- (x^{2}) - (- 9 x) - 1 \cdot - 10 = 0$

Step 4.15.1.4

Factor $- 1$ out of $- (x^{2}) - (- 9 x)$ .

$- (x^{2} - 9 x) - 1 \cdot - 10 = 0$

Step 4.15.1.5

Factor $- 1$ out of $- (x^{2} - 9 x) - 1 (- 10)$ .

$- (x^{2} - 9 x - 10) = 0$

Step 4.15.2

Factor.

Tap for more steps...

Step 4.15.2.1

Factor $x^{2} - 9 x - 10$ using the AC method.

Tap for more steps...

Step 4.15.2.1.1

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 $- 10$ and whose sum is $- 9$ .

$- 10, 1$

Step 4.15.2.1.2

Write the factored form using these integers.

$- ((x - 10) (x + 1)) = 0$

Step 4.15.2.2

Remove unnecessary parentheses.

$- (x - 10) (x + 1) = 0$

Step 4.16

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

$x - 10 = 0$

$x + 1 = 0$

Step 4.17

Set $x - 10$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.17.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 4.17.2

Add $10$ to both sides of the equation.

$x = 10$

Step 4.18

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

Tap for more steps...

Step 4.18.1

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

$x + 1 = 0$

Step 4.18.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4.19

The final solution is all the values that make $- (x - 10) (x + 1) = 0$ true.

$x = 10, - 1$

Step 4.20

The complete solution is the result of both the positive and negative portions of the solution.

$x = 10, 1, - 1$

Step 5

Exclude the solutions that do not make $| x - 10 | = x^{2} - 10 x$ true.

$x = 10, - 1$