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

In summary: Since all of the square roots of 64 are whole numbers, it would follow that all of the solutions to the original equation would be rational numbers. To do this, you would need to find the square root of the discriminant, which is (8m-12). Since 8m-12 is a whole number, the square root of 8m-12 is a whole number as well. This means that all of the solutions to the equation would be rational numbers.
  • #1
danago
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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:

[tex]12x^2 + 8mx + (4m-3) = 0[/tex]

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:
[tex]\Delta = (8m)^2 - 4(12)(4m-3)[/tex]

simplified it:
[tex]\Delta = 64m^2 - 192m + 144[/tex]

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.

If anybody has any idea about if i should have gone about this another way, or if i was right, please post :) all comments greatly appreciated.

Thanks,
Dan.
 
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  • #2
Try factoring the discriminant. Then use the quadratic formula to show that all answers are rational.
 
  • #3
danago said:
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:

[tex]12x^2 + 8mx + (4m-3) = 0[/tex]

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:
[tex]\Delta = (8m)^2 - 4(12)(4m-3)[/tex]

simplified it:
[tex]\Delta = 64m^2 - 192m + 144[/tex]
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:
[tex]\frac{-8m \pm \sqrt{\Delta}}{24}[/tex] is a rational number, right?
Note that:
[tex]\Delta = 64m ^ 2 - 192m + 144 = (8m - 12) ^ 2[/tex].
Can you go from here? :)
 

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

How do you solve for m in a quadratic equation?

To solve for m in a quadratic equation, you must first rearrange the equation into the standard form of ax^2 + bx + c = 0. In this case, the equation is 12x^2 + 8mx + (4m-3) = 0. Once the equation is in standard form, you can use the quadratic formula, which is m = (-b ± √(b^2-4ac)) / 2a, to solve for m.

What is the significance of the quadratic formula in solving for m?

The quadratic formula is a powerful tool in solving quadratic equations, as it provides a formula for finding the roots of any quadratic equation, including those with complex solutions. It is derived from the standard form of a quadratic equation and can be used to solve for any variable, including m.

Can you use factoring to find m in a quadratic equation?

Yes, factoring can also be used to find m in a quadratic equation. However, it may not always be possible to factor the quadratic equation, especially if the value of m is not a perfect square. In these cases, the quadratic formula may be a better method for solving for m.

How many solutions can a quadratic equation have for m?

A quadratic equation can have two solutions for m, one solution, or no real solutions. This is determined by the discriminant, which is the value inside the square root in the quadratic formula. If the discriminant is positive, there will be two real solutions. If the discriminant is zero, there will be one real solution. And if the discriminant is negative, there will be no real solutions.

Are there any real-world applications for finding m in a quadratic equation?

Yes, quadratic equations are commonly used in fields such as physics, engineering, and economics to model real-world situations. For example, finding the optimal value of m in a quadratic equation can be used to determine the maximum profit or minimum cost of a business. Additionally, many natural phenomena can be modeled using quadratic equations, such as the trajectory of a projectile or the growth of a population.

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