Basic Math Examples

$3 z^{2} - 6 z = 72$

Step 1

Subtract $72$ from both sides of the equation.

$3 z^{2} - 6 z - 72 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $3$ out of $3 z^{2} - 6 z - 72$ .

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Step 2.1.1

Factor $3$ out of $3 z^{2}$ .

$3 (z^{2}) - 6 z - 72 = 0$

Step 2.1.2

Factor $3$ out of $- 6 z$ .

$3 (z^{2}) + 3 (- 2 z) - 72 = 0$

Step 2.1.3

Factor $3$ out of $- 72$ .

$3 z^{2} + 3 (- 2 z) + 3 \cdot - 24 = 0$

Step 2.1.4

Factor $3$ out of $3 z^{2} + 3 (- 2 z)$ .

$3 (z^{2} - 2 z) + 3 \cdot - 24 = 0$

Step 2.1.5

Factor $3$ out of $3 (z^{2} - 2 z) + 3 \cdot - 24$ .

$3 (z^{2} - 2 z - 24) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor $z^{2} - 2 z - 24$ using the AC method.

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Step 2.2.1.1

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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Step 2.2.1.2

Write the factored form using these integers.

$3 ((z - 6) (z + 4)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$3 (z - 6) (z + 4) = 0$

Step 3

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

$z - 6 = 0$

$z + 4 = 0$

Step 4

Set $z - 6$ equal to $0$ and solve for $z$ .

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Step 4.1

Set $z - 6$ equal to $0$ .

$z - 6 = 0$

Step 4.2

Add $6$ to both sides of the equation.

$z = 6$

Step 5

Set $z + 4$ equal to $0$ and solve for $z$ .

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Step 5.1

Set $z + 4$ equal to $0$ .

$z + 4 = 0$

Step 5.2

Subtract $4$ from both sides of the equation.

$z = - 4$

Step 6

The final solution is all the values that make $3 (z - 6) (z + 4) = 0$ true.

$z = 6, - 4$