Find All Integers k Such That the Trinomial Can Be Factored 4x^2-kx+49

$4 x^{2} - k x + 49$

Step 1

Find the values of $a$ and $c$ in the trinomial $4 x^{2} - k x + 49$ with the format $a x^{2} + k x + c$ .

$a = 4$

$c = 49$

Step 2

For the trinomial $4 x^{2} - k x + 49$ , find the value of $a \cdot c$ .

$a \cdot c = 196$

Step 3

To find all possible values of $k$ , first find the factors of $a \cdot c$ $196$ . Once a factor is found, add it to its corresponding factor to get a possible value for $k$ . The factors for $196$ are all numbers between $- 196$ and $196$ , which divide $196$ evenly.

Check numbers between $- 196$ and $196$

Step 4

Calculate the factors of $196$ . Add corresponding factors to get all possible $k$ values.

Since $196$ divided by $- 196$ is the whole number $- 1$ , $- 196$ and $- 1$ are factors of $196$ .

$- 196$ and $- 1$ are factors

Add the factors $- 196$ and $- 1$ together. Add $- 197$ to the list of possible $k$ values.

$k = - 197$

Since $196$ divided by $- 98$ is the whole number $- 2$ , $- 98$ and $- 2$ are factors of $196$ .

$- 98$ and $- 2$ are factors

Add the factors $- 98$ and $- 2$ together. Add $- 100$ to the list of possible $k$ values.

$k = - 197, - 100$

Since $196$ divided by $- 49$ is the whole number $- 4$ , $- 49$ and $- 4$ are factors of $196$ .

$- 49$ and $- 4$ are factors

Add the factors $- 49$ and $- 4$ together. Add $- 53$ to the list of possible $k$ values.

$k = - 197, - 100, - 53$

Since $196$ divided by $- 28$ is the whole number $- 7$ , $- 28$ and $- 7$ are factors of $196$ .

$- 28$ and $- 7$ are factors

Add the factors $- 28$ and $- 7$ together. Add $- 35$ to the list of possible $k$ values.

$k = - 197, - 100, - 53, - 35$

Since $196$ divided by $- 14$ is the whole number $- 14$ , $- 14$ and $- 14$ are factors of $196$ .

$- 14$ and $- 14$ are factors

Add the factors $- 14$ and $- 14$ together. Add $- 28$ to the list of possible $k$ values.

$k = - 197, - 100, - 53, - 35, - 28$

Since $196$ divided by $1$ is the whole number $196$ , $1$ and $196$ are factors of $196$ .

$1$ and $196$ are factors

Add the factors $1$ and $196$ together. Add $197$ to the list of possible $k$ values.

$k = - 197, - 100, - 53, - 35, - 28, 197$

Since $196$ divided by $2$ is the whole number $98$ , $2$ and $98$ are factors of $196$ .

$2$ and $98$ are factors

Add the factors $2$ and $98$ together. Add $100$ to the list of possible $k$ values.

$k = - 197, - 100, - 53, - 35, - 28, 197, 100$

Since $196$ divided by $4$ is the whole number $49$ , $4$ and $49$ are factors of $196$ .

$4$ and $49$ are factors

Add the factors $4$ and $49$ together. Add $53$ to the list of possible $k$ values.

$k = - 197, - 100, - 53, - 35, - 28, 197, 100, 53$

Since $196$ divided by $7$ is the whole number $28$ , $7$ and $28$ are factors of $196$ .

$7$ and $28$ are factors

Add the factors $7$ and $28$ together. Add $35$ to the list of possible $k$ values.

$k = - 197, - 100, - 53, - 35, - 28, 197, 100, 53, 35$

Since $196$ divided by $14$ is the whole number $14$ , $14$ and $14$ are factors of $196$ .

$14$ and $14$ are factors

Add the factors $14$ and $14$ together. Add $28$ to the list of possible $k$ values.

$k = - 197, - 100, - 53, - 35, - 28, 197, 100, 53, 35, 28$

Find All Integers k Such That the Trinomial Can Be Factored 4x^2-kx+49

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