Find the function when given x

Question

X = -5 and X=2 find the function and provide the proof. I have no idea of where to start with this, please help!

Answer 1

At a first glance, the problem doesn't have enough information.

1. When x = -5 and x = 2, what is f(-5) and f(2)?
2. What kind of function is f(x)?

If those are f(-5) = f(2) = 0, then we know they are zeros of the function (or roots if it is a polynomial).

If f(x) is a quadratic, then f(x) = k(x + 5)(x - 2) = k(x² - x*2 + 5*x - 10) = k(x² + 3x - 10), where k is a real number.

If f(x) is a monic quadratic (that is, the coefficient of the highest power is 1), then f(x) = x² + 3x - 10.

Answer 2

If x = -5 and x = 2 are pieces of given information, then we can assume that the solutions for x refer to zeroes of a function. One way to express zeroes is through linear factors. For x = -5, we know that -5 + 5 = 0. By replacing the "-5" with the x that is assigned to that value, we end up with x + 5. The (x + 5) becomes our linear factor associated with x = -5. Similarly for x = 2 we know that 2 - 2 = 0, so that means x - 2 = 0 and we end up with our other linear factor. Since we can end up with 0 by multiplying 0 by any possible number, we then can combine our two linear factors through multiplication. Thus we end up with (x + 5)(x - 2). Then using the distributive property(the upcoming * will define multiplication), we do x * x = x^2, 5 * x = 5x, x * -2 = -2x, and 5 * -2 = -10. Adding all of our products provides us with x^2 + 3x - 10. Sure enough if you plug in -5 or 2 for x, you will end up with an output of 0 from the function you just came up with.

Answer 3

It looks like x= -5 and x= 2 are solutions/roots (when f(x)=0) of a function, so (x+5) and (x-2) are factors of the function f(x)

So f(x) = (x+5)(x-2) then use the distributive property of multiplication.

f(x) = x•x -x•2 + 5•x -5•2 complete multiplication in each term

f(x) =x² -2x +5x -10 combine like terms

f(x) = x² +3x -10

Answer 4

I think the problem is asking for a function such that for f(x)=0, x = -5 and x=2.

f(x) = (x+5)(x-2) = x² - 2x + 5x -10 = x²+3x-10 using foil.

f(-5)=25-15-10 = 0

f(2)=4+6-10 = 0