# Finding m in quadratic equation 12x^2 + 8mx + (4m-3) = 0 d

1. Apr 5, 2006

### danago

Hey. Today i had a test on quadratics and discriminants. I think i did fairly well, but i am a bit confused about one of the questions i had in it.

We were given the following quadratic equation:

$$12x^2 + 8mx + (4m-3) = 0$$

What we had to do was prove that for any integer value of m, the equation would have rational solutions.

What i did was first take the discriminant of the equation:
$$\Delta = (8m)^2 - 4(12)(4m-3)$$

simplified it:
$$\Delta = 64m^2 - 192m + 144$$

From that, i then created a table of values. I made a table of values for m (-5 to 5), the discriminant, then the square root of the discriminant. Since all of the square roots were whole numbers, i could have used that as a reason why all the solutions would be rational, but its not really proving that all values for m will follow the rule. It just shows that 10 of my chosen values work.

from here i wasnt really sure what to do. I noted that the discriminant of the original equation was a quadratic function itself, so i graphed it, and noticed that for every integer value of x, its corrosponding y value will be a perfect square number. I wrote about this observation, and am just hoping its close enough to what i should have done.

Thanks,
Dan.

2. Apr 5, 2006

### nocturnal

Try factoring the discriminant. Then use the quadratic formula to show that all answers are rational.

3. Apr 5, 2006

### VietDao29

So far so good.
Do you know what are rational numbers? They are numbers that can be expressed in a form of a fraction p / q, where p, and q are whole numbers (integers).
To prove that the solutions are rational for any integer value of m, you should prove that:
$$\frac{-8m \pm \sqrt{\Delta}}{24}$$ is a rational number, right?
Note that:
$$\Delta = 64m ^ 2 - 192m + 144 = (8m - 12) ^ 2$$.
Can you go from here? :)