Finite Math Examples

$6 - | 2 x^{2} - 8 x - 1 | = - 3$

Step 1

Move all terms not containing $| 2 x^{2} - 8 x - 1 |$ to the right side of the equation.

Tap for more steps...

Step 1.1

Subtract $6$ from both sides of the equation.

$- | 2 x^{2} - 8 x - 1 | = - 3 - 6$

Step 1.2

Subtract $6$ from $- 3$ .

$- | 2 x^{2} - 8 x - 1 | = - 9$

Step 2

Divide each term in $- | 2 x^{2} - 8 x - 1 | = - 9$ by $- 1$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $- | 2 x^{2} - 8 x - 1 | = - 9$ by $- 1$ .

$\frac{- | 2 x^{2} - 8 x - 1 |}{- 1} = \frac{- 9}{- 1}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{| 2 x^{2} - 8 x - 1 |}{1} = \frac{- 9}{- 1}$

Step 2.2.2

Divide $| 2 x^{2} - 8 x - 1 |$ by $1$ .

$| 2 x^{2} - 8 x - 1 | = \frac{- 9}{- 1}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Divide $- 9$ by $- 1$ .

$| 2 x^{2} - 8 x - 1 | = 9$

Step 3

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

$2 x^{2} - 8 x - 1 = \pm 9$

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.

$2 x^{2} - 8 x - 1 = 9$

Step 4.2

Subtract $9$ from both sides of the equation.

$2 x^{2} - 8 x - 1 - 9 = 0$

Step 4.3

Subtract $9$ from $- 1$ .

$2 x^{2} - 8 x - 10 = 0$

Step 4.4

Factor the left side of the equation.

Tap for more steps...

Step 4.4.1

Factor $2$ out of $2 x^{2} - 8 x - 10$ .

Tap for more steps...

Step 4.4.1.1

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

$2 (x^{2}) - 8 x - 10 = 0$

Step 4.4.1.2

Factor $2$ out of $- 8 x$ .

$2 (x^{2}) + 2 (- 4 x) - 10 = 0$

Step 4.4.1.3

Factor $2$ out of $- 10$ .

$2 x^{2} + 2 (- 4 x) + 2 \cdot - 5 = 0$

Step 4.4.1.4

Factor $2$ out of $2 x^{2} + 2 (- 4 x)$ .

$2 (x^{2} - 4 x) + 2 \cdot - 5 = 0$

Step 4.4.1.5

Factor $2$ out of $2 (x^{2} - 4 x) + 2 \cdot - 5$ .

$2 (x^{2} - 4 x - 5) = 0$

Step 4.4.2

Factor.

Tap for more steps...

Step 4.4.2.1

Factor $x^{2} - 4 x - 5$ using the AC method.

Tap for more steps...

Step 4.4.2.1.1

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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 4.4.2.1.2

Write the factored form using these integers.

$2 ((x - 5) (x + 1)) = 0$

Step 4.4.2.2

Remove unnecessary parentheses.

$2 (x - 5) (x + 1) = 0$

Step 4.5

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

$x - 5 = 0$

$x + 1 = 0$

Step 4.6

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

Tap for more steps...

Step 4.6.1

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

$x - 5 = 0$

Step 4.6.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.7

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

Tap for more steps...

Step 4.7.1

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

$x + 1 = 0$

Step 4.7.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4.8

The final solution is all the values that make $2 (x - 5) (x + 1) = 0$ true.

$x = 5, - 1$

Step 4.9

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

$2 x^{2} - 8 x - 1 = - 9$

Step 4.10

Add $9$ to both sides of the equation.

$2 x^{2} - 8 x - 1 + 9 = 0$

Step 4.11

Add $- 1$ and $9$ .

$2 x^{2} - 8 x + 8 = 0$

Step 4.12

Factor the left side of the equation.

Tap for more steps...

Step 4.12.1

Factor $2$ out of $2 x^{2} - 8 x + 8$ .

Tap for more steps...

Step 4.12.1.1

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

$2 (x^{2}) - 8 x + 8 = 0$

Step 4.12.1.2

Factor $2$ out of $- 8 x$ .

$2 (x^{2}) + 2 (- 4 x) + 8 = 0$

Step 4.12.1.3

Factor $2$ out of $8$ .

$2 x^{2} + 2 (- 4 x) + 2 \cdot 4 = 0$

Step 4.12.1.4

Factor $2$ out of $2 x^{2} + 2 (- 4 x)$ .

$2 (x^{2} - 4 x) + 2 \cdot 4 = 0$

Step 4.12.1.5

Factor $2$ out of $2 (x^{2} - 4 x) + 2 \cdot 4$ .

$2 (x^{2} - 4 x + 4) = 0$

Step 4.12.2

Factor using the perfect square rule.

Tap for more steps...

Step 4.12.2.1

Rewrite $4$ as $2^{2}$ .

$2 (x^{2} - 4 x + 2^{2}) = 0$

Step 4.12.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 4.12.2.3

Rewrite the polynomial.

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

Step 4.12.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 4.13

Divide each term in $2 {(x - 2)}^{2} = 0$ by $2$ and simplify.

Tap for more steps...

Step 4.13.1

Divide each term in $2 {(x - 2)}^{2} = 0$ by $2$ .

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

Step 4.13.2

Simplify the left side.

Tap for more steps...

Step 4.13.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.13.2.1.1

Cancel the common factor.

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

Step 4.13.2.1.2

Divide ${(x - 2)}^{2}$ by $1$ .

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

Step 4.13.3

Simplify the right side.

Tap for more steps...

Step 4.13.3.1

Divide $0$ by $2$ .

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

Step 4.14

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.15

Add $2$ to both sides of the equation.

$x = 2$

Step 4.16

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

$x = 5, - 1, 2$