In The Given Equation, $5x^2 - Ax + 64 = 0$, $a$ Is A Constant.For Which Of The Following Values Of \$a$[/tex\] Will The Equation Have More Than One Real Solution?A. 20 B. $-39$ C. 5 D. 8
Introduction
Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on solving the quadratic equation $5x^2 - ax + 64 = 0$, where $a$ is a constant. Our goal is to determine the values of $a$ that will result in more than one real solution.
Understanding Quadratic Equations
A quadratic equation can be written in the general form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The solutions to a quadratic equation can be found using various methods, including factoring, completing the square, and the quadratic formula. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
The Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It provides a general solution to the equation, and it can be used to find the roots of the equation. However, the quadratic formula only works if the discriminant, $b^2 - 4ac$, is non-negative. If the discriminant is negative, the equation has no real solutions.
The Discriminant
The discriminant is a critical component of the quadratic formula. It determines the nature of the solutions to the equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.
Applying the Quadratic Formula to the Given Equation
Now that we have a good understanding of quadratic equations and the quadratic formula, let's apply it to the given equation $5x^2 - ax + 64 = 0$. We can use the quadratic formula to find the solutions to the equation. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Finding the Discriminant
To find the discriminant, we need to calculate $b^2 - 4ac$. In this case, $a = 5$, $b = -a$, and $c = 64$. Plugging these values into the formula, we get:
Determining the Values of a
Now that we have the discriminant, we can determine the values of $a$ that will result in more than one real solution. We know that the discriminant must be positive for the equation to have two distinct real solutions. Therefore, we can set up the inequality $a^2 - 1280 > 0$.
Solving the Inequality
To solve the inequality, we can add 1280 to both sides, resulting in $a^2 > 1280$. Taking the square root of both sides, we get $|a| > \sqrt{1280}$. Simplifying the right-hand side, we get $|a| > 35.77$.
Finding the Values of a
Now that we have the inequality $|a| > 35.77$, we can find the values of $a$ that satisfy this inequality. We know that $a$ must be greater than 35.77 or less than -35.77.
Conclusion
In conclusion, the values of $a$ that will result in more than one real solution to the equation $5x^2 - ax + 64 = 0$ are $a > 35.77$ or $a < -35.77$. Therefore, the correct answer is not among the options provided. However, we can use the options to estimate the correct answer.
Estimating the Correct Answer
Let's examine the options provided. We know that $a$ must be greater than 35.77 or less than -35.77. Looking at the options, we can see that $a = 20$ is less than 35.77, and $a = -39$ is less than -35.77. Therefore, the correct answer is likely to be one of these two options.
Final Answer
Based on our analysis, we can conclude that the correct answer is $a = -39$. This value of $a$ will result in more than one real solution to the equation $5x^2 - ax + 64 = 0$.
References
- [1] "Quadratic Equations" by Math Open Reference
- [2] "Quadratic Formula" by Wolfram MathWorld
- [3] "Discriminant" by Khan Academy
Additional Resources
- [1] "Quadratic Equations" by MIT OpenCourseWare
- [2] "Quadratic Formula" by Purplemath
- [3] "Discriminant" by Mathway
Quadratic Equations Q&A ==========================
Frequently Asked Questions
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. It can be written in the general form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.
Q: How do I solve a quadratic equation?
A: There are several methods to solve a quadratic equation, including factoring, completing the square, and the quadratic formula. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Q: What is the discriminant?
A: The discriminant is a critical component of the quadratic formula. It determines the nature of the solutions to the equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.
Q: How do I find the discriminant?
A: To find the discriminant, you need to calculate $b^2 - 4ac$. This involves plugging in the values of $a$, $b$, and $c$ into the formula.
Q: What are the values of a that will result in more than one real solution?
A: The values of $a$ that will result in more than one real solution are $a > 35.77$ or $a < -35.77$.
Q: How do I determine the values of a that will result in more than one real solution?
A: To determine the values of $a$ that will result in more than one real solution, you need to solve the inequality $a^2 - 1280 > 0$. This involves adding 1280 to both sides, resulting in $a^2 > 1280$, and then taking the square root of both sides.
Q: What is the correct answer to the problem?
A: The correct answer to the problem is $a = -39$.
Q: What are some additional resources for learning about quadratic equations?
A: Some additional resources for learning about quadratic equations include:
- [1] "Quadratic Equations" by MIT OpenCourseWare
- [2] "Quadratic Formula" by Purplemath
- [3] "Discriminant" by Mathway
Q: What are some common mistakes to avoid when solving quadratic equations?
A: Some common mistakes to avoid when solving quadratic equations include:
- Not checking the discriminant before using the quadratic formula
- Not simplifying the expression under the square root
- Not checking for extraneous solutions
Q: How do I check for extraneous solutions?
A: To check for extraneous solutions, you need to plug the solutions back into the original equation and check if they are true. If they are not true, then they are extraneous solutions.
Q: What are some real-world applications of quadratic equations?
A: Some real-world applications of quadratic equations include:
- Modeling the trajectory of a projectile
- Finding the maximum or minimum value of a function
- Solving problems involving optimization
Q: How do I use quadratic equations to model real-world problems?
A: To use quadratic equations to model real-world problems, you need to identify the variables and the relationships between them. You can then use the quadratic formula to solve for the unknown variable.
Q: What are some tips for solving quadratic equations?
A: Some tips for solving quadratic equations include:
- Checking the discriminant before using the quadratic formula
- Simplifying the expression under the square root
- Checking for extraneous solutions
- Using the quadratic formula to solve for the unknown variable
Q: How do I practice solving quadratic equations?
A: To practice solving quadratic equations, you can try solving problems on your own or using online resources such as Khan Academy or Mathway. You can also try solving problems with a partner or in a study group.