Find the product of all constants t such that the quadratic x^2 tx - 9 can be factored in the form (x a) (x b), where a and b are integers.

product of the constants P will be

P = 12

Step-by-step explanation:

the quadratic equation

F = x² + t*x - 9

has as solution

a and b = [-t ± √ (t² - 4*1 * (-9)) ] / 2]

then

a - b = - t/2

a = b - t/2

since b is an integer, then t/2 should be an integer, then t=2*n, where n is any integer

also

a and b = [-t ± √ (t² - 4*1 * (-9)) ] / 2] = [-2*n ± √ (4*n²+36) ] / 2 = - n ± n √ (1+9 / n²]

since n are integers, then √ (1+9 / n²] should be and integer and therefore

9 / n² should be an integer. Then the possible values of n are

n=1 and n=3

therefore the possible values of t are

t₁=2*1 = 2

t₂=2*3 = 6

the product of the constants P will be

P=t₁*t₂ = 12