Question 1209974: For how many integer values of a does the equation
x^2 + ax + 12a = 5x + 8
have integer solutions for x?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Alright, let's solve this problem step-by-step.
**1. Rearrange the Equation**
First, let's rearrange the given equation into a standard quadratic form:
$$x^2 + ax + 12a = 5x + 8$$
$$x^2 + (a - 5)x + (12a - 8) = 0$$
**2. Apply the Quadratic Formula**
For the equation to have integer solutions for *x*, the discriminant must be a perfect square. The quadratic formula is:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
In our case, *a* = 1, *b* = (a - 5), and *c* = (12a - 8). So:
$$x = \frac{-(a - 5) \pm \sqrt{(a - 5)^2 - 4(1)(12a - 8)}}{2(1)}$$
$$x = \frac{5 - a \pm \sqrt{a^2 - 10a + 25 - 48a + 32}}{2}$$
$$x = \frac{5 - a \pm \sqrt{a^2 - 58a + 57}}{2}$$
**3. Analyze the Discriminant**
For *x* to be an integer, the discriminant (the expression inside the square root) must be a perfect square. Let's call the discriminant *D*:
$$D = a^2 - 58a + 57$$
We need to find integer values of *a* such that *D* is a perfect square. Let *D* = *k*², where *k* is a non-negative integer.
$$a^2 - 58a + 57 = k^2$$
Complete the square for the *a* terms:
$$(a - 29)^2 - 29^2 + 57 = k^2$$
$$(a - 29)^2 - 841 + 57 = k^2$$
$$(a - 29)^2 - 784 = k^2$$
$$(a - 29)^2 - k^2 = 784$$
$$(a - 29 - k)(a - 29 + k) = 784$$
**4. Factor 784**
We need to find pairs of factors of 784. The prime factorization of 784 is 2⁴ * 7².
The pairs of factors of 784 are:
(1, 784), (2, 392), (4, 196), (7, 112), (8, 98), (14, 56), (16, 49), (28, 28)
We also need to consider the negative pairs:
(-1, -784), (-2, -392), (-4, -196), (-7, -112), (-8, -98), (-14, -56), (-16, -49), (-28, -28)
**5. Solve for 'a'**
Let's set up a system of equations for each pair of factors.
Let:
* a - 29 - k = m
* a - 29 + k = n
Then, m * n = 784. Adding the two equations, we get:
2a - 58 = m + n
a = (m + n + 58) / 2
Since *a* must be an integer, *m + n + 58* must be even. This means *m* and *n* must have the same parity (both even or both odd). Since 784 is even, all pairs of factors will have the same parity.
Now, we calculate the values of *a* for each pair of factors:
* (1, 784): a = (1 + 784 + 58) / 2 = 421.5 (not an integer)
* (2, 392): a = (2 + 392 + 58) / 2 = 226
* (4, 196): a = (4 + 196 + 58) / 2 = 129
* (7, 112): a = (7 + 112 + 58) / 2 = 88.5 (not an integer)
* (8, 98): a = (8 + 98 + 58) / 2 = 82
* (14, 56): a = (14 + 56 + 58) / 2 = 64
* (16, 49): a = (16 + 49 + 58) / 2 = 61.5 (not an integer)
* (28, 28): a = (28 + 28 + 58) / 2 = 57
* (-1, -784): a = (-1 - 784 + 58) / 2 = -363.5 (not an integer)
* (-2, -392): a = (-2 - 392 + 58) / 2 = -168
* (-4, -196): a = (-4 - 196 + 58) / 2 = -71
* (-7, -112): a = (-7 - 112 + 58) / 2 = -30.5 (not an integer)
* (-8, -98): a = (-8 - 98 + 58) / 2 = -24
* (-14, -56): a = (-14 - 56 + 58) / 2 = -6
* (-16, -49): a = (-16 - 49 + 58) / 2 = -3.5 (not an integer)
* (-28, -28): a = (-28 - 28 + 58) / 2 = 1
The integer values of *a* are: 226, 129, 82, 64, 57, -168, -71, -24, -6, 1.
There are 10 integer values of *a*.
**Final Answer**
There are 10 integer values of *a*.
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