Find All Integers k Such That the Trinomial Can Be Factored 100x^2+kx+64

$100 x^{2} + k x + 64$

Step 1

Find the values of $a$ and $c$ in the trinomial $100 x^{2} + k x + 64$ with the format $a x^{2} + k x + c$ .

$a = 100$

$c = 64$

Step 2

For the trinomial $100 x^{2} + k x + 64$ , find the value of $a \cdot c$ .

$a \cdot c = 6400$

Step 3

To find all possible values of $k$ , first find the factors of $a \cdot c$ $6400$ . Once a factor is found, add it to its corresponding factor to get a possible value for $k$ . The factors for $6400$ are all numbers between $- 6400$ and $6400$ , which divide $6400$ evenly.

Check numbers between $- 6400$ and $6400$

Step 4

Calculate the factors of $6400$ . Add corresponding factors to get all possible $k$ values.

Since $6400$ divided by $- 6400$ is the whole number $- 1$ , $- 6400$ and $- 1$ are factors of $6400$ .

$- 6400$ and $- 1$ are factors

Add the factors $- 6400$ and $- 1$ together. Add $- 6401$ to the list of possible $k$ values.

$k = - 6401$

Since $6400$ divided by $- 3200$ is the whole number $- 2$ , $- 3200$ and $- 2$ are factors of $6400$ .

$- 3200$ and $- 2$ are factors

Add the factors $- 3200$ and $- 2$ together. Add $- 3202$ to the list of possible $k$ values.

$k = - 6401, - 3202$

Since $6400$ divided by $- 1600$ is the whole number $- 4$ , $- 1600$ and $- 4$ are factors of $6400$ .

$- 1600$ and $- 4$ are factors

Add the factors $- 1600$ and $- 4$ together. Add $- 1604$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604$

Since $6400$ divided by $- 1280$ is the whole number $- 5$ , $- 1280$ and $- 5$ are factors of $6400$ .

$- 1280$ and $- 5$ are factors

Add the factors $- 1280$ and $- 5$ together. Add $- 1285$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285$

Since $6400$ divided by $- 800$ is the whole number $- 8$ , $- 800$ and $- 8$ are factors of $6400$ .

$- 800$ and $- 8$ are factors

Add the factors $- 800$ and $- 8$ together. Add $- 808$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808$

Since $6400$ divided by $- 640$ is the whole number $- 10$ , $- 640$ and $- 10$ are factors of $6400$ .

$- 640$ and $- 10$ are factors

Add the factors $- 640$ and $- 10$ together. Add $- 650$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650$

Since $6400$ divided by $- 400$ is the whole number $- 16$ , $- 400$ and $- 16$ are factors of $6400$ .

$- 400$ and $- 16$ are factors

Add the factors $- 400$ and $- 16$ together. Add $- 416$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416$

Since $6400$ divided by $- 320$ is the whole number $- 20$ , $- 320$ and $- 20$ are factors of $6400$ .

$- 320$ and $- 20$ are factors

Add the factors $- 320$ and $- 20$ together. Add $- 340$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340$

Since $6400$ divided by $- 256$ is the whole number $- 25$ , $- 256$ and $- 25$ are factors of $6400$ .

$- 256$ and $- 25$ are factors

Add the factors $- 256$ and $- 25$ together. Add $- 281$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281$

Since $6400$ divided by $- 200$ is the whole number $- 32$ , $- 200$ and $- 32$ are factors of $6400$ .

$- 200$ and $- 32$ are factors

Add the factors $- 200$ and $- 32$ together. Add $- 232$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232$

Since $6400$ divided by $- 160$ is the whole number $- 40$ , $- 160$ and $- 40$ are factors of $6400$ .

$- 160$ and $- 40$ are factors

Add the factors $- 160$ and $- 40$ together. Add $- 200$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200$

Since $6400$ divided by $- 128$ is the whole number $- 50$ , $- 128$ and $- 50$ are factors of $6400$ .

$- 128$ and $- 50$ are factors

Add the factors $- 128$ and $- 50$ together. Add $- 178$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178$

Since $6400$ divided by $- 100$ is the whole number $- 64$ , $- 100$ and $- 64$ are factors of $6400$ .

$- 100$ and $- 64$ are factors

Add the factors $- 100$ and $- 64$ together. Add $- 164$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164$

Since $6400$ divided by $- 80$ is the whole number $- 80$ , $- 80$ and $- 80$ are factors of $6400$ .

$- 80$ and $- 80$ are factors

Add the factors $- 80$ and $- 80$ together. Add $- 160$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160$

Since $6400$ divided by $1$ is the whole number $6400$ , $1$ and $6400$ are factors of $6400$ .

$1$ and $6400$ are factors

Add the factors $1$ and $6400$ together. Add $6401$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401$

Since $6400$ divided by $2$ is the whole number $3200$ , $2$ and $3200$ are factors of $6400$ .

$2$ and $3200$ are factors

Add the factors $2$ and $3200$ together. Add $3202$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202$

Since $6400$ divided by $4$ is the whole number $1600$ , $4$ and $1600$ are factors of $6400$ .

$4$ and $1600$ are factors

Add the factors $4$ and $1600$ together. Add $1604$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604$

Since $6400$ divided by $5$ is the whole number $1280$ , $5$ and $1280$ are factors of $6400$ .

$5$ and $1280$ are factors

Add the factors $5$ and $1280$ together. Add $1285$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604, 1285$

Since $6400$ divided by $8$ is the whole number $800$ , $8$ and $800$ are factors of $6400$ .

$8$ and $800$ are factors

Add the factors $8$ and $800$ together. Add $808$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604, 1285, 808$

Since $6400$ divided by $10$ is the whole number $640$ , $10$ and $640$ are factors of $6400$ .

$10$ and $640$ are factors

Add the factors $10$ and $640$ together. Add $650$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604, 1285, 808, 650$

Since $6400$ divided by $16$ is the whole number $400$ , $16$ and $400$ are factors of $6400$ .

$16$ and $400$ are factors

Add the factors $16$ and $400$ together. Add $416$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604, 1285, 808, 650, 416$

Since $6400$ divided by $20$ is the whole number $320$ , $20$ and $320$ are factors of $6400$ .

$20$ and $320$ are factors

Add the factors $20$ and $320$ together. Add $340$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604, 1285, 808, 650, 416, 340$

Since $6400$ divided by $25$ is the whole number $256$ , $25$ and $256$ are factors of $6400$ .

$25$ and $256$ are factors

Add the factors $25$ and $256$ together. Add $281$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604, 1285, 808, 650, 416, 340, 281$

Since $6400$ divided by $32$ is the whole number $200$ , $32$ and $200$ are factors of $6400$ .

$32$ and $200$ are factors

Add the factors $32$ and $200$ together. Add $232$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604, 1285, 808, 650, 416, 340, 281, 232$

Since $6400$ divided by $40$ is the whole number $160$ , $40$ and $160$ are factors of $6400$ .

$40$ and $160$ are factors

Add the factors $40$ and $160$ together. Add $200$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604, 1285, 808, 650, 416, 340, 281, 232, 200$

Since $6400$ divided by $50$ is the whole number $128$ , $50$ and $128$ are factors of $6400$ .

$50$ and $128$ are factors

Add the factors $50$ and $128$ together. Add $178$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604, 1285, 808, 650, 416, 340, 281, 232, 200, 178$

Since $6400$ divided by $64$ is the whole number $100$ , $64$ and $100$ are factors of $6400$ .

$64$ and $100$ are factors

Add the factors $64$ and $100$ together. Add $164$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604, 1285, 808, 650, 416, 340, 281, 232, 200, 178, 164$

Since $6400$ divided by $80$ is the whole number $80$ , $80$ and $80$ are factors of $6400$ .

$80$ and $80$ are factors

Add the factors $80$ and $80$ together. Add $160$ to the list of possible $k$ values.

$k = - 6401, - 3202, - 1604, - 1285, - 808, - 650, - 416, - 340, - 281, - 232, - 200, - 178, - 164, - 160, 6401, 3202, 1604, 1285, 808, 650, 416, 340, 281, 232, 200, 178, 164, 160$

Find All Integers k Such That the Trinomial Can Be Factored 100x^2+kx+64

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