Algebra Examples

$z (z + 7) = 30$

Step 1

Simplify $z (z + 7)$ .

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Step 1.1

Apply the distributive property.

$z \cdot z + z \cdot 7 = 30$

Step 1.2

Simplify the expression.

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Step 1.2.1

Multiply $z$ by $z$ .

$z^{2} + z \cdot 7 = 30$

Step 1.2.2

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

$z^{2} + 7 z = 30$

Step 2

Subtract $30$ from both sides of the equation.

$z^{2} + 7 z - 30 = 0$

Step 3

Factor $z^{2} + 7 z - 30$ using the AC method.

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Step 3.1

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 $- 30$ and whose sum is $7$ .

$- 3, 10$

Step 3.2

Write the factored form using these integers.

$(z - 3) (z + 10) = 0$

Step 4

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

$z - 3 = 0$

$z + 10 = 0$

Step 5

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

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Step 5.1

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

$z - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$z = 3$

Step 6

Set $z + 10$ equal to $0$ and solve for $z$ .

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Step 6.1

Set $z + 10$ equal to $0$ .

$z + 10 = 0$

Step 6.2

Subtract $10$ from both sides of the equation.

$z = - 10$

Step 7

The final solution is all the values that make $(z - 3) (z + 10) = 0$ true.

$z = 3, - 10$