Find The Values Of \[$ K \$\] For Which The Equation \[$(k-3)x^2 + (k+3)x + K+3 = 0\$\] Has A Repeated Root.

Introduction

In algebra, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is ${ax^2 + bx + c = 0}$, where ${a}$, ${b}$, and ${c}$ are constants, and ${x}$ is the variable. In this article, we will focus on the quadratic equation ${(k-3)x^2 + (k+3)x + k+3 = 0}$, where ${k}$ is a constant. Our goal is to find the values of ${k}$ for which this equation has a repeated root.

What is a Repeated Root?

A repeated root, also known as a double root, is a root that occurs twice in a quadratic equation. In other words, if a quadratic equation has a repeated root, it means that the equation can be factored as ${(x - r)^2 = 0}$, where ${r}$ is the repeated root.

The Condition for a Repeated Root

For a quadratic equation to have a repeated root, the discriminant must be equal to zero. The discriminant is the expression under the square root in the quadratic formula, which is given by ${b^2 - 4ac}$. In our case, the discriminant is ${(k+3)^2 - 4(k-3)(k+3)}$.

Simplifying the Discriminant

To simplify the discriminant, we can expand the squares and multiply out the terms:

${(k+3)^2 - 4(k-3)(k+3) = k^2 + 6k + 9 - 4(k^2 - 2k - 9)}$

Expanding the second term, we get:

${k^2 + 6k + 9 - 4k^2 + 8k + 36}$

Combining like terms, we get:

${-3k^2 + 14k + 45}$

Setting the Discriminant Equal to Zero

For the quadratic equation to have a repeated root, the discriminant must be equal to zero. Therefore, we set the simplified discriminant equal to zero:

${-3k^2 + 14k + 45 = 0}$

Solving the Quadratic Equation

To solve this quadratic equation, we can use the quadratic formula:

${k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$

In this case, ${a = -3}$, ${b = 14}$, and ${c = 45}$. Plugging these values into the formula, we get:

${k = \frac{-14 \pm \sqrt{14^2 - 4(-3)(45)}}{2(-3)}}$

Simplifying the expression under the square root, we get:

${k = \frac{-14 \pm \sqrt{196 + 540}}{-6}}$

${k = \frac{-14 \pm \sqrt{736}}{-6}}$

${k = \frac{-14 \pm 8\sqrt{23}}{-6}}$

Simplifying the Solutions

To simplify the solutions, we can divide both the numerator and the denominator by their greatest common divisor, which is 2:

${k = \frac{-7 \pm 4\sqrt{23}}{-3}}$

Conclusion

In this article, we have found the values of ${k}$ for which the quadratic equation ${(k-3)x^2 + (k+3)x + k+3 = 0}$ has a repeated root. The solutions are given by:

${k = \frac{-7 \pm 4\sqrt{23}}{-3}}$

These values of ${k}$ will result in a repeated root for the given quadratic equation.

Applications of Repeated Roots

Repeated roots have many applications in mathematics and science. For example, in physics, repeated roots can be used to model the motion of an object that oscillates at a constant frequency. In engineering, repeated roots can be used to design filters that remove unwanted frequencies from a signal.

Future Research Directions

There are many open research directions in the study of repeated roots. For example, researchers could investigate the properties of repeated roots in more complex polynomial equations. They could also explore the applications of repeated roots in machine learning and data analysis.

References

• [1] "Quadratic Equations" by Michael Artin, Springer-Verlag, 2010.
• [2] "Algebra" by Michael Artin, Springer-Verlag, 2011.
• [3] "The Quadratic Formula" by Math Open Reference, 2020.

Note: The references provided are for illustrative purposes only and are not actual references used in this article.

Introduction

In our previous article, we discussed the concept of repeated roots in quadratic equations and found the values of ${k}$ for which the equation ${(k-3)x^2 + (k+3)x + k+3 = 0}$ has a repeated root. In this article, we will answer some frequently asked questions about repeated roots.

Q: What is a repeated root?

A: A repeated root, also known as a double root, is a root that occurs twice in a quadratic equation. In other words, if a quadratic equation has a repeated root, it means that the equation can be factored as ${(x - r)^2 = 0}$, where ${r}$ is the repeated root.

Q: How do I find the values of k for which a quadratic equation has a repeated root?

A: To find the values of ${k}$ for which a quadratic equation has a repeated root, you need to set the discriminant equal to zero and solve for ${k}$. The discriminant is the expression under the square root in the quadratic formula, which is given by ${b^2 - 4ac}$.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by ${b^2 - 4ac}$. It is used to determine the nature of the roots of a quadratic equation.

Q: How do I simplify the discriminant?

A: To simplify the discriminant, you can expand the squares and multiply out the terms. You can also combine like terms to get a simpler expression.

Q: What are the applications of repeated roots?

A: Repeated roots have many applications in mathematics and science. For example, in physics, repeated roots can be used to model the motion of an object that oscillates at a constant frequency. In engineering, repeated roots can be used to design filters that remove unwanted frequencies from a signal.

Q: Can repeated roots occur in higher-degree polynomial equations?

A: Yes, repeated roots can occur in higher-degree polynomial equations. However, the process of finding the values of the coefficients for which the equation has a repeated root is more complex and requires the use of advanced mathematical techniques.

Q: How do I determine if a quadratic equation has a repeated root?

A: To determine if a quadratic equation has a repeated root, you can use the quadratic formula and check if the discriminant is equal to zero. If the discriminant is equal to zero, then the equation has a repeated root.

Q: Can repeated roots be used in machine learning and data analysis?

A: Yes, repeated roots can be used in machine learning and data analysis. For example, repeated roots can be used to model the behavior of complex systems and to identify patterns in data.

Q: What are some common mistakes to avoid when working with repeated roots?

A: Some common mistakes to avoid when working with repeated roots include:

• Not setting the discriminant equal to zero when finding the values of the coefficients for which the equation has a repeated root.
• Not simplifying the discriminant correctly.
• Not using the correct formula for the quadratic equation.
• Not checking if the equation has a repeated root before solving for the roots.

Conclusion

In this article, we have answered some frequently asked questions about repeated roots. We hope that this article has been helpful in clarifying the concept of repeated roots and providing a better understanding of how to work with them.

References

• [1] "Quadratic Equations" by Michael Artin, Springer-Verlag, 2010.
• [2] "Algebra" by Michael Artin, Springer-Verlag, 2011.
• [3] "The Quadratic Formula" by Math Open Reference, 2020.

Note: The references provided are for illustrative purposes only and are not actual references used in this article.