The Roots Of A Quadratic Equation Are Given By $x = \frac{-3 \pm \sqrt{13-2k}}{4}$. Calculate The Value(s) Of $k$ For Which The Roots Are Equal.

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. The roots of a quadratic equation are the values of $x$ that satisfy the equation. In this article, we will explore the roots of a quadratic equation given by $x = \frac{-3 \pm \sqrt{13-2k}}{4}$ and calculate the value(s) of $k$ for which the roots are equal.

The Quadratic Formula

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

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

In our case, the quadratic equation is $x = \frac{-3 \pm \sqrt{13-2k}}{4}$. Comparing this with the quadratic formula, we can see that $a = \frac{1}{4}$, $b = -\frac{3}{2}$, and $c = \frac{13-2k}{4}$.

Equal Roots

For the roots of a quadratic equation to be equal, the discriminant must be zero. The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$. In our case, the discriminant is $13-2k$.

Setting the Discriminant to Zero

To find the value(s) of $k$ for which the roots are equal, we need to set the discriminant to zero and solve for $k$. This gives us the equation:

$13-2k = 0$

Solving for $k$

To solve for $k$, we need to isolate $k$ on one side of the equation. This gives us:

$-2k = -13$

Dividing both sides by $-2$ gives us:

$k = \frac{13}{2}$

Conclusion

In this article, we explored the roots of a quadratic equation given by $x = \frac{-3 \pm \sqrt{13-2k}}{4}$ and calculated the value(s) of $k$ for which the roots are equal. We used the quadratic formula to find the solutions to the quadratic equation and set the discriminant to zero to find the value of $k$. The final answer is $k = \frac{13}{2}$.

Additional Information

• The quadratic formula is a powerful tool for solving quadratic equations.
• The discriminant is an important concept in algebra, as it determines the nature of the roots of a quadratic equation.
• In this article, we used the quadratic formula to find the solutions to a quadratic equation and set the discriminant to zero to find the value of $k$.

References

• [1] "Quadratic Formula" by Math Open Reference. Math Open Reference, 2022.
• [2] "Discriminant" by Khan Academy. Khan Academy, 2022.

Related Topics

• Quadratic Equations
• Algebra
• Mathematics

Tags

• Quadratic Equation
• Algebra
• Mathematics
• Quadratic Formula
• Discriminant
  

Introduction

In our previous article, we explored the roots of a quadratic equation given by $x = \frac{-3 \pm \sqrt{13-2k}}{4}$ and calculated the value(s) of $k$ for which the roots are equal. In this article, we will provide a Q&A guide to help you better understand the concepts and calculations involved.

Q: What is a quadratic equation?

A: 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.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

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

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$. It determines the nature of the roots of a quadratic equation.

Q: How do I find the value(s) of $k$ for which the roots are equal?

A: To find the value(s) of $k$ for which the roots are equal, you need to set the discriminant to zero and solve for $k$. This gives you the equation:

$13-2k = 0$

Q: How do I solve for $k$?

A: To solve for $k$, you need to isolate $k$ on one side of the equation. This gives you:

$-2k = -13$

Dividing both sides by $-2$ gives you:

$k = \frac{13}{2}$

Q: What is the significance of the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It provides a general solution to any quadratic equation, regardless of the values of $a$, $b$, and $c$.

Q: What is the importance of the discriminant?

A: The discriminant is an important concept in algebra, as it determines the nature of the roots of a quadratic equation. If the discriminant is positive, the roots are real and distinct. If the discriminant is zero, the roots are real and equal. If the discriminant is negative, the roots are complex.

Q: Can you provide more examples of quadratic equations?

A: Yes, here are a few examples of quadratic equations:

• $x^2 + 4x + 4 = 0$
• $x^2 - 6x + 8 = 0$
• $x^2 + 2x - 3 = 0$

Q: How do I apply the quadratic formula to these examples?

A: To apply the quadratic formula to these examples, you need to identify the values of $a$, $b$, and $c$ in each equation. Then, you can plug these values into the quadratic formula to find the solutions.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

• Not identifying the values of $a$, $b$, and $c$ correctly
• Not plugging the correct values into the quadratic formula
• Not simplifying the expression under the square root
• Not checking the discriminant to determine the nature of the roots

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts and calculations involved in finding the roots of a quadratic equation. We covered topics such as the quadratic formula, the discriminant, and common mistakes to avoid. We hope this guide has been helpful in your studies of algebra and mathematics.

Additional Information

• The quadratic formula is a powerful tool for solving quadratic equations.
• The discriminant is an important concept in algebra, as it determines the nature of the roots of a quadratic equation.
• In this article, we provided a Q&A guide to help you better understand the concepts and calculations involved in finding the roots of a quadratic equation.

References

• [1] "Quadratic Formula" by Math Open Reference. Math Open Reference, 2022.
• [2] "Discriminant" by Khan Academy. Khan Academy, 2022.

Related Topics

• Quadratic Equations
• Algebra
• Mathematics

Tags

• Quadratic Equation
• Algebra
• Mathematics
• Quadratic Formula
• Discriminant