Use The Discriminant To Determine All Values Of { K $}$ Which Would Result In The Equation $ K X^2 - 12 X + 12 = 0 $ Having Equal Roots.
Introduction
In algebra, the discriminant is a value that can be calculated from the coefficients of a quadratic equation. It is used to determine the nature of the roots of the equation, whether they are real and distinct, real and equal, or complex. In this article, we will explore how to use the discriminant to determine all values of k which would result in the equation $ k x^2 - 12 x + 12 = 0 $ having equal roots.
What is the Discriminant?
The discriminant of a quadratic equation in the form $ ax^2 + bx + c = 0 $ is given by the formula $ D = b^2 - 4ac $. It is a value that can be calculated from the coefficients of the equation. The discriminant is used to determine the nature of the roots of the equation.
Nature of Roots
The nature of the roots of a quadratic equation can be determined by the value of the discriminant. If the discriminant is positive, the equation has two real and distinct roots. If the discriminant is zero, the equation has two real and equal roots. If the discriminant is negative, the equation has two complex roots.
Equal Roots
In this article, we are interested in finding the values of k which would result in the equation $ k x^2 - 12 x + 12 = 0 $ having equal roots. This means that the discriminant of the equation must be equal to zero.
Calculating the Discriminant
To calculate the discriminant of the equation $ k x^2 - 12 x + 12 = 0 $, we can use the formula $ D = b^2 - 4ac $. In this case, a = k, b = -12, and c = 12. Plugging these values into the formula, we get:
$ D = (-12)^2 - 4(k)(12) $
$ D = 144 - 48k $
Setting the Discriminant Equal to Zero
Since we are interested in finding the values of k which would result in the equation having equal roots, we set the discriminant equal to zero:
$ 144 - 48k = 0 $
Solving for k
To solve for k, we can add 48k to both sides of the equation:
$ 144 = 48k $
Then, we can divide both sides of the equation by 48:
$ k = \frac{144}{48} $
$ k = 3 $
Conclusion
In this article, we used the discriminant to determine all values of k which would result in the equation $ k x^2 - 12 x + 12 = 0 $ having equal roots. We calculated the discriminant of the equation and set it equal to zero. Then, we solved for k and found that k = 3.
Example
Let's consider an example to illustrate how to use the discriminant to determine the values of k which would result in the equation having equal roots.
Suppose we have the equation $ 2x^2 - 6x + 3 = 0 $. We can calculate the discriminant of the equation using the formula $ D = b^2 - 4ac $. In this case, a = 2, b = -6, and c = 3. Plugging these values into the formula, we get:
$ D = (-6)^2 - 4(2)(3) $
$ D = 36 - 24 $
$ D = 12 $
Since the discriminant is positive, the equation has two real and distinct roots.
Tips and Tricks
Here are some tips and tricks to keep in mind when using the discriminant to determine the values of k which would result in the equation having equal roots:
- Make sure to calculate the discriminant correctly using the formula $ D = b^2 - 4ac $.
- Set the discriminant equal to zero to find the values of k which would result in the equation having equal roots.
- Solve for k by adding 48k to both sides of the equation and then dividing both sides of the equation by 48.
- Use the discriminant to determine the nature of the roots of the equation, whether they are real and distinct, real and equal, or complex.
Common Mistakes
Here are some common mistakes to avoid when using the discriminant to determine the values of k which would result in the equation having equal roots:
- Calculating the discriminant incorrectly using the formula $ D = b^2 - 4ac $.
- Not setting the discriminant equal to zero to find the values of k which would result in the equation having equal roots.
- Not solving for k correctly by adding 48k to both sides of the equation and then dividing both sides of the equation by 48.
- Not using the discriminant to determine the nature of the roots of the equation, whether they are real and distinct, real and equal, or complex.
Conclusion
In conclusion, the discriminant is a powerful tool used to determine the nature of the roots of a quadratic equation. By calculating the discriminant and setting it equal to zero, we can find the values of k which would result in the equation having equal roots. We can use the discriminant to determine the nature of the roots of the equation, whether they are real and distinct, real and equal, or complex. By following the tips and tricks outlined in this article, we can avoid common mistakes and use the discriminant effectively to determine the values of k which would result in the equation having equal roots.
Q: What is the discriminant, and how is it used in quadratic equations?
A: The discriminant is a value that can be calculated from the coefficients of a quadratic equation. It is used to determine the nature of the roots of the equation, whether they are real and distinct, real and equal, or complex.
Q: How do I calculate the discriminant of a quadratic equation?
A: To calculate the discriminant of a quadratic equation in the form $ ax^2 + bx + c = 0 $, you can use the formula $ D = b^2 - 4ac $. Simply plug in the values of a, b, and c into the formula, and you will get the discriminant.
Q: What does the discriminant tell me about the roots of the equation?
A: The discriminant tells you the nature of the roots of the equation. If the discriminant is positive, the equation has two real and distinct roots. If the discriminant is zero, the equation has two real and equal roots. If the discriminant is negative, the equation has two complex roots.
Q: How do I use the discriminant to determine the values of k which would result in the equation having equal roots?
A: To use the discriminant to determine the values of k which would result in the equation having equal roots, you need to set the discriminant equal to zero and solve for k. This will give you the values of k which would result in the equation having equal roots.
Q: What are some common mistakes to avoid when using the discriminant to determine the values of k?
A: Some common mistakes to avoid when using the discriminant to determine the values of k include calculating the discriminant incorrectly, not setting the discriminant equal to zero, and not solving for k correctly.
Q: Can I use the discriminant to determine the values of k which would result in the equation having complex roots?
A: Yes, you can use the discriminant to determine the values of k which would result in the equation having complex roots. If the discriminant is negative, the equation has two complex roots.
Q: How do I determine the values of k which would result in the equation having real and distinct roots?
A: To determine the values of k which would result in the equation having real and distinct roots, you need to set the discriminant greater than zero and solve for k. This will give you the values of k which would result in the equation having real and distinct roots.
Q: Can I use the discriminant to determine the values of k which would result in the equation having no real roots?
A: Yes, you can use the discriminant to determine the values of k which would result in the equation having no real roots. If the discriminant is negative, the equation has no real roots.
Q: How do I use the discriminant to determine the values of k which would result in the equation having one real root and one complex root?
A: To use the discriminant to determine the values of k which would result in the equation having one real root and one complex root, you need to set the discriminant equal to zero and solve for k. This will give you the values of k which would result in the equation having one real root and one complex root.
Q: Can I use the discriminant to determine the values of k which would result in the equation having two real and distinct roots and two complex roots?
A: No, you cannot use the discriminant to determine the values of k which would result in the equation having two real and distinct roots and two complex roots. The discriminant can only be used to determine the nature of the roots of the equation, whether they are real and distinct, real and equal, or complex.
Q: How do I use the discriminant to determine the values of k which would result in the equation having two real and distinct roots and one real root and one complex root?
A: To use the discriminant to determine the values of k which would result in the equation having two real and distinct roots and one real root and one complex root, you need to set the discriminant greater than zero and solve for k. This will give you the values of k which would result in the equation having two real and distinct roots, and the values of k which would result in the equation having one real root and one complex root.
Q: Can I use the discriminant to determine the values of k which would result in the equation having two real and distinct roots, two real and equal roots, and two complex roots?
A: Yes, you can use the discriminant to determine the values of k which would result in the equation having two real and distinct roots, two real and equal roots, and two complex roots. By setting the discriminant equal to zero, greater than zero, and less than zero, you can determine the values of k which would result in the equation having two real and distinct roots, two real and equal roots, and two complex roots.
Conclusion
In conclusion, the discriminant is a powerful tool used to determine the nature of the roots of a quadratic equation. By calculating the discriminant and using it to determine the values of k which would result in the equation having equal roots, real and distinct roots, complex roots, and no real roots, you can gain a deeper understanding of the behavior of quadratic equations.