Basic Math Examples

$| t^{2} - 9 | - | 2 t - 6 | = 0$

Step 1

Subtract $| t^{2} - 9 |$ from both sides of the equation.

$- | 2 t - 6 | = - | t^{2} - 9 |$

Step 2

Divide each term in $- | 2 t - 6 | = - | t^{2} - 9 |$ by $- 1$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $- | 2 t - 6 | = - | t^{2} - 9 |$ by $- 1$ .

$\frac{- | 2 t - 6 |}{- 1} = \frac{- | t^{2} - 9 |}{- 1}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{| 2 t - 6 |}{1} = \frac{- | t^{2} - 9 |}{- 1}$

Step 2.2.2

Divide $| 2 t - 6 |$ by $1$ .

$| 2 t - 6 | = \frac{- | t^{2} - 9 |}{- 1}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Dividing two negative values results in a positive value.

$| 2 t - 6 | = \frac{| t^{2} - 9 |}{1}$

Step 2.3.2

Divide $| t^{2} - 9 |$ by $1$ .

$| 2 t - 6 | = | t^{2} - 9 |$

Step 3

Rewrite the absolute value equation as four equations without absolute value bars.

$2 t - 6 = t^{2} - 9$

$2 t - 6 = - (t^{2} - 9)$

$- (2 t - 6) = t^{2} - 9$

$- (2 t - 6) = - (t^{2} - 9)$

Step 4

After simplifying, there are only two unique equations to be solved.

$2 t - 6 = t^{2} - 9$

$2 t - 6 = - (t^{2} - 9)$

Step 5

Solve $2 t - 6 = t^{2} - 9$ for $t$ .

Tap for more steps...

Step 5.1

Subtract $t^{2}$ from both sides of the equation.

$2 t - 6 - t^{2} = - 9$

Step 5.2

Add $9$ to both sides of the equation.

$2 t - 6 - t^{2} + 9 = 0$

Step 5.3

Add $- 6$ and $9$ .

$2 t - t^{2} + 3 = 0$

Step 5.4

Factor the left side of the equation.

Tap for more steps...

Step 5.4.1

Factor $- 1$ out of $2 t - t^{2} + 3$ .

Tap for more steps...

Step 5.4.1.1

Reorder $2 t$ and $- t^{2}$ .

$- t^{2} + 2 t + 3 = 0$

Step 5.4.1.2

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

$- (t^{2}) + 2 t + 3 = 0$

Step 5.4.1.3

Factor $- 1$ out of $2 t$ .

$- (t^{2}) - (- 2 t) + 3 = 0$

Step 5.4.1.4

Rewrite $3$ as $- 1 (- 3)$ .

$- (t^{2}) - (- 2 t) - 1 \cdot - 3 = 0$

Step 5.4.1.5

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

$- (t^{2} - 2 t) - 1 \cdot - 3 = 0$

Step 5.4.1.6

Factor $- 1$ out of $- (t^{2} - 2 t) - 1 (- 3)$ .

$- (t^{2} - 2 t - 3) = 0$

Step 5.4.2

Factor.

Tap for more steps...

Step 5.4.2.1

Factor $t^{2} - 2 t - 3$ using the AC method.

Tap for more steps...

Step 5.4.2.1.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 5.4.2.1.2

Write the factored form using these integers.

$- ((t - 3) (t + 1)) = 0$

Step 5.4.2.2

Remove unnecessary parentheses.

$- (t - 3) (t + 1) = 0$

Step 5.5

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

$t - 3 = 0$

$t + 1 = 0$

Step 5.6

Set $t - 3$ equal to $0$ and solve for $t$ .

Tap for more steps...

Step 5.6.1

Set $t - 3$ equal to $0$ .

$t - 3 = 0$

Step 5.6.2

Add $3$ to both sides of the equation.

$t = 3$

Step 5.7

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

Tap for more steps...

Step 5.7.1

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

$t + 1 = 0$

Step 5.7.2

Subtract $1$ from both sides of the equation.

$t = - 1$

Step 5.8

The final solution is all the values that make $- (t - 3) (t + 1) = 0$ true.

$t = 3, - 1$

Step 6

Solve $2 t - 6 = - (t^{2} - 9)$ for $t$ .

Tap for more steps...

Step 6.1

Simplify $- (t^{2} - 9)$ .

Tap for more steps...

Step 6.1.1

Rewrite.

$2 t - 6 = 0 + 0 - (t^{2} - 9)$

Step 6.1.2

Simplify by adding zeros.

$2 t - 6 = - (t^{2} - 9)$

Step 6.1.3

Apply the distributive property.

$2 t - 6 = - t^{2} - - 9$

Step 6.1.4

Multiply $- 1$ by $- 9$ .

$2 t - 6 = - t^{2} + 9$

Step 6.2

Add $t^{2}$ to both sides of the equation.

$2 t - 6 + t^{2} = 9$

Step 6.3

Subtract $9$ from both sides of the equation.

$2 t - 6 + t^{2} - 9 = 0$

Step 6.4

Subtract $9$ from $- 6$ .

$2 t + t^{2} - 15 = 0$

Step 6.5

Factor the left side of the equation.

Tap for more steps...

Step 6.5.1

Let $u = t$ . Substitute $u$ for all occurrences of $t$ .

$2 u + u^{2} - 15 = 0$

Step 6.5.2

Factor $2 u + u^{2} - 15$ using the AC method.

Tap for more steps...

Step 6.5.2.1

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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 6.5.2.2

Write the factored form using these integers.

$(u - 3) (u + 5) = 0$

Step 6.5.3

Replace all occurrences of $u$ with $t$ .

$(t - 3) (t + 5) = 0$

Step 6.6

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

$t - 3 = 0$

$t + 5 = 0$

Step 6.7

Set $t - 3$ equal to $0$ and solve for $t$ .

Tap for more steps...

Step 6.7.1

Set $t - 3$ equal to $0$ .

$t - 3 = 0$

Step 6.7.2

Add $3$ to both sides of the equation.

$t = 3$

Step 6.8

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

Tap for more steps...

Step 6.8.1

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

$t + 5 = 0$

Step 6.8.2

Subtract $5$ from both sides of the equation.

$t = - 5$

Step 6.9

The final solution is all the values that make $(t - 3) (t + 5) = 0$ true.

$t = 3, - 5$

Step 7

List all of the solutions.

$t = 3, - 1, 3, - 5$

Step 8