Finite Math Examples

$t (7 + t) = 4 (t + 7)$

Step 1

Simplify $t (7 + t)$ .

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Step 1.1

Rewrite.

$0 + 0 + t (7 + t) = 4 (t + 7)$

Step 1.2

Simplify by adding zeros.

$t (7 + t) = 4 (t + 7)$

Step 1.3

Apply the distributive property.

$t \cdot 7 + t \cdot t = 4 (t + 7)$

Step 1.4

Simplify the expression.

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Step 1.4.1

Move $7$ to the left of $t$ .

$7 \cdot t + t \cdot t = 4 (t + 7)$

Step 1.4.2

Multiply $t$ by $t$ .

$7 t + t^{2} = 4 (t + 7)$

Step 2

Simplify $4 (t + 7)$ .

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Step 2.1

Apply the distributive property.

$7 t + t^{2} = 4 t + 4 \cdot 7$

Step 2.2

Multiply $4$ by $7$ .

$7 t + t^{2} = 4 t + 28$

Step 3

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

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Step 3.1

Subtract $4 t$ from both sides of the equation.

$7 t + t^{2} - 4 t = 28$

Step 3.2

Subtract $4 t$ from $7 t$ .

$t^{2} + 3 t = 28$

Step 4

Subtract $28$ from both sides of the equation.

$t^{2} + 3 t - 28 = 0$

Step 5

Factor $t^{2} + 3 t - 28$ using the AC method.

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Step 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 $- 28$ and whose sum is $3$ .

$- 4, 7$

Step 5.2

Write the factored form using these integers.

$(t - 4) (t + 7) = 0$

Step 6

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

$t - 4 = 0$

$t + 7 = 0$

Step 7

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

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Step 7.1

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

$t - 4 = 0$

Step 7.2

Add $4$ to both sides of the equation.

$t = 4$

Step 8

Set $t + 7$ equal to $0$ and solve for $t$ .

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Step 8.1

Set $t + 7$ equal to $0$ .

$t + 7 = 0$

Step 8.2

Subtract $7$ from both sides of the equation.

$t = - 7$

Step 9

The final solution is all the values that make $(t - 4) (t + 7) = 0$ true.

$t = 4, - 7$