Pre-Algebra Examples

$10^{2} = z (21 + z)$

Step 1

Rewrite the equation as $z (21 + z) = 10^{2}$ .

$z (21 + z) = 10^{2}$

Step 2

Simplify $z (21 + z)$ .

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Step 2.1

Apply the distributive property.

$z \cdot 21 + z \cdot z = 10^{2}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $21$ to the left of $z$ .

$21 \cdot z + z \cdot z = 10^{2}$

Step 2.2.2

Multiply $z$ by $z$ .

$21 z + z^{2} = 10^{2}$

Step 3

Raise $10$ to the power of $2$ .

$21 z + z^{2} = 100$

Step 4

Subtract $100$ from both sides of the equation.

$21 z + z^{2} - 100 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

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

$21 u + u^{2} - 100 = 0$

Step 5.2

Factor $21 u + u^{2} - 100$ using the AC method.

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Step 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 $- 100$ and whose sum is $21$ .

$- 4, 25$

Step 5.2.2

Write the factored form using these integers.

$(u - 4) (u + 25) = 0$

Step 5.3

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

$(z - 4) (z + 25) = 0$

Step 6

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

$z - 4 = 0$

$z + 25 = 0$

Step 7

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

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Step 7.1

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

$z - 4 = 0$

Step 7.2

Add $4$ to both sides of the equation.

$z = 4$

Step 8

Set $z + 25$ equal to $0$ and solve for $z$ .

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Step 8.1

Set $z + 25$ equal to $0$ .

$z + 25 = 0$

Step 8.2

Subtract $25$ from both sides of the equation.

$z = - 25$

Step 9

The final solution is all the values that make $(z - 4) (z + 25) = 0$ true.

$z = 4, - 25$