In the equation x^2 + mx + n = 0, m and n are integers. The only possible value for x is -3. What is the value of m?
If I substitute -3 and solve for m, I can get many values for m based on what I pick for n. How can the answer only be 6?
If the only possible value for x is 3, then the function must factor into
(x+3)(x+3) = x^2+6x+9
When they say the only possible value, they mean the only value which makes the equation true. Naturally, x can have many other values, but in those cases, the value on the left is not zero.
If they had said that there was only a single root of -3, you probably would have had no trouble, eh?
To find the value of m given that the only possible value for x is -3, we can substitute x = -3 into the equation x^2 + mx + n = 0 and solve for m.
Substituting x = -3, we get:
(-3)^2 + m(-3) + n = 0
9 - 3m + n = 0
Since m and n are integers, this equation implies that n = -9 + 3m.
Now, we know that the only possible value for x is -3. This means that the equation has only one solution, which implies that it must be a perfect square trinomial. In order for a trinomial to be a perfect square, the coefficient of x must be twice the square root of the constant term.
In this case, the constant term is n. So, to satisfy the condition that the coefficient of x is twice the square root of n, we need m = 2√n.
Since n = -9 + 3m, we can substitute this expression for n into the equation above:
m = 2√(-9 + 3m)
Simplifying this equation:
m = 2√(-9 + 3m)
m/2 = √(-9 + 3m)
Square both sides to remove the square root:
(m/2)^2 = -9 + 3m
Expanding and simplifying:
m^2/4 = -9 + 3m
m^2 = -36 + 12m
m^2 - 12m + 36 = 0
(m - 6)^2 = 0
Taking the square root of both sides:
m - 6 = 0
m = 6
Therefore, the value of m when the only possible value for x is -3 is m = 6.
Note that in the original equation, we did not substitute any specific value for n. The equation only guarantees that if x = -3 is the only possible solution, then m = 6. The value of n can vary, but it does not affect the value of m.