Precalculus Examples

$z^{2} + 24 z + 95 = 0$

Step 1

Factor $z^{2} + 24 z + 95$ using the AC method.

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Step 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 $95$ and whose sum is $24$ .

$5, 19$

Step 1.2

Write the factored form using these integers.

$(z + 5) (z + 19) = 0$

Step 2

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

$z + 5 = 0$

$z + 19 = 0$

Step 3

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

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Step 3.1

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

$z + 5 = 0$

Step 3.2

Subtract $5$ from both sides of the equation.

$z = - 5$

Step 4

Set $z + 19$ equal to $0$ and solve for $z$ .

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Step 4.1

Set $z + 19$ equal to $0$ .

$z + 19 = 0$

Step 4.2

Subtract $19$ from both sides of the equation.

$z = - 19$

Step 5

The final solution is all the values that make $(z + 5) (z + 19) = 0$ true.

$z = - 5, - 19$