Algebra Examples

$z - 5 = z^{2} - 25$

Step 1

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

$z - 5 - z^{2} = - 25$

Step 2

Add $25$ to both sides of the equation.

$z - 5 - z^{2} + 25 = 0$

Step 3

Add $- 5$ and $25$ .

$z - z^{2} + 20 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $- 1$ out of $z - z^{2} + 20$ .

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Step 4.1.1

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

$- z^{2} + z + 20 = 0$

Step 4.1.2

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

$- (z^{2}) + z + 20 = 0$

Step 4.1.3

Factor $- 1$ out of $z$ .

$- (z^{2}) - 1 (- z) + 20 = 0$

Step 4.1.4

Rewrite $20$ as $- 1 (- 20)$ .

$- (z^{2}) - 1 (- z) - 1 \cdot - 20 = 0$

Step 4.1.5

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

$- (z^{2} - z) - 1 \cdot - 20 = 0$

Step 4.1.6

Factor $- 1$ out of $- (z^{2} - z) - 1 (- 20)$ .

$- (z^{2} - z - 20) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor $z^{2} - z - 20$ using the AC method.

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Step 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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 4.2.1.2

Write the factored form using these integers.

$- ((z - 5) (z + 4)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$- (z - 5) (z + 4) = 0$

Step 5

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

$z - 5 = 0$

$z + 4 = 0$

Step 6

Set $z - 5$ equal to $0$ and solve for $z$ .

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Step 6.1

Set $z - 5$ equal to $0$ .

$z - 5 = 0$

Step 6.2

Add $5$ to both sides of the equation.

$z = 5$

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

Subtract $4$ from both sides of the equation.

$z = - 4$

Step 8

The final solution is all the values that make $- (z - 5) (z + 4) = 0$ true.

$z = 5, - 4$