Precalculus Examples

$25 x^{2} - k x + 4$

Step 1

Find the values of $a$ and $c$ in the trinomial $25 x^{2} - k x + 4$ with the format $a x^{2} + k x + c$ .

$a = 25$

$c = 4$

Step 2

For the trinomial $25 x^{2} - k x + 4$ , find the value of $a \cdot c$ .

$a \cdot c = 100$

Step 3

To find all possible values of $k$ , first find the factors of $a \cdot c$ $100$ . Once a factor is found, add it to its corresponding factor to get a possible value for $k$ . The factors for $100$ are all numbers between $- 100$ and $100$ , which divide $100$ evenly.

Check numbers between $- 100$ and $100$

Step 4

Calculate the factors of $100$ . Add corresponding factors to get all possible $k$ values.

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Step 4.1

Since $100$ divided by $- 100$ is the whole number $- 1$ , $- 100$ and $- 1$ are factors of $100$ .

$- 100$ and $- 1$ are factors

Step 4.2

Add the factors $- 100$ and $- 1$ together. Add $- 101$ to the list of possible $k$ values.

$k = - 101$

Step 4.3

Since $100$ divided by $- 50$ is the whole number $- 2$ , $- 50$ and $- 2$ are factors of $100$ .

$- 50$ and $- 2$ are factors

Step 4.4

Add the factors $- 50$ and $- 2$ together. Add $- 52$ to the list of possible $k$ values.

$k = - 101, - 52$

Step 4.5

Since $100$ divided by $- 25$ is the whole number $- 4$ , $- 25$ and $- 4$ are factors of $100$ .

$- 25$ and $- 4$ are factors

Step 4.6

Add the factors $- 25$ and $- 4$ together. Add $- 29$ to the list of possible $k$ values.

$k = - 101, - 52, - 29$

Step 4.7

Since $100$ divided by $- 20$ is the whole number $- 5$ , $- 20$ and $- 5$ are factors of $100$ .

$- 20$ and $- 5$ are factors

Step 4.8

Add the factors $- 20$ and $- 5$ together. Add $- 25$ to the list of possible $k$ values.

$k = - 101, - 52, - 29, - 25$

Step 4.9

Since $100$ divided by $- 10$ is the whole number $- 10$ , $- 10$ and $- 10$ are factors of $100$ .

$- 10$ and $- 10$ are factors

Step 4.10

Add the factors $- 10$ and $- 10$ together. Add $- 20$ to the list of possible $k$ values.

$k = - 101, - 52, - 29, - 25, - 20$

Step 4.11

Since $100$ divided by $1$ is the whole number $100$ , $1$ and $100$ are factors of $100$ .

$1$ and $100$ are factors

Step 4.12

Add the factors $1$ and $100$ together. Add $101$ to the list of possible $k$ values.

$k = - 101, - 52, - 29, - 25, - 20, 101$

Step 4.13

Since $100$ divided by $2$ is the whole number $50$ , $2$ and $50$ are factors of $100$ .

$2$ and $50$ are factors

Step 4.14

Add the factors $2$ and $50$ together. Add $52$ to the list of possible $k$ values.

$k = - 101, - 52, - 29, - 25, - 20, 101, 52$

Step 4.15

Since $100$ divided by $4$ is the whole number $25$ , $4$ and $25$ are factors of $100$ .

$4$ and $25$ are factors

Step 4.16

Add the factors $4$ and $25$ together. Add $29$ to the list of possible $k$ values.

$k = - 101, - 52, - 29, - 25, - 20, 101, 52, 29$

Step 4.17

Since $100$ divided by $5$ is the whole number $20$ , $5$ and $20$ are factors of $100$ .

$5$ and $20$ are factors

Step 4.18

Add the factors $5$ and $20$ together. Add $25$ to the list of possible $k$ values.

$k = - 101, - 52, - 29, - 25, - 20, 101, 52, 29, 25$

Step 4.19

Since $100$ divided by $10$ is the whole number $10$ , $10$ and $10$ are factors of $100$ .

$10$ and $10$ are factors

Step 4.20

Add the factors $10$ and $10$ together. Add $20$ to the list of possible $k$ values.

$k = - 101, - 52, - 29, - 25, - 20, 101, 52, 29, 25, 20$