Enter The Correct Answer In The Box.What Are The Solutions Of This Quadratic Equation? X 2 = 16 X − 65 X^2 = 16x - 65 X 2 = 16 X − 65 Substitute The Values Of A A A And B B B To Complete The Solutions.$[ \begin{array}{l} x = A + B I \ x = A - B

Mar 11, 2025 by ADMIN 245 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will delve into the world of quadratic equations and explore the solutions to the equation $x^2 = 16x - 65$ . We will also discuss the importance of quadratic equations in real-world applications and provide a step-by-step guide on how to solve them.

What are Quadratic Equations?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation. The quadratic formula can be used to find the solutions to any quadratic equation, regardless of whether it can be factored or not.

Solving the Quadratic Equation $x^2 = 16x - 65$

To solve the quadratic equation $x^2 = 16x - 65$ , we need to rewrite it in the standard form $ax^2 + bx + c = 0$ . We can do this by subtracting $16x$ from both sides and adding $65$ to both sides:

$x^2 - 16x + 65 = 0$

Now, we can use the quadratic formula to find the solutions to this equation. We have:

$a = 1$ , $b = -16$ , and $c = 65$

Plugging these values into the quadratic formula, we get:

$x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(65)}}{2(1)}$

Simplifying this expression, we get:

$x = \frac{16 \pm \sqrt{256 - 260}}{2}$

$x = \frac{16 \pm \sqrt{-4}}{2}$

$x = \frac{16 \pm 2i}{2}$

$x = 8 \pm i$

Therefore, the solutions to the quadratic equation $x^2 = 16x - 65$ are $x = 8 + i$ and $x = 8 - i$ .

Completing the Square

Completing the square is another method for solving quadratic equations. It involves rewriting the quadratic equation in a form that allows us to easily identify the solutions. To complete the square, we need to add and subtract a constant term to the equation. This constant term is called the "square root of the constant term".

For example, consider the quadratic equation $x^2 + 6x + 8 = 0$ . To complete the square, we need to add and subtract $9$ to the equation:

$x^2 + 6x + 9 - 9 + 8 = 0$

$(x + 3)^2 - 1 = 0$

$(x + 3)^2 = 1$

$x + 3 = \pm 1$

$x = -3 \pm 1$

$x = -4$ or $x = -2$

Therefore, the solutions to the quadratic equation $x^2 + 6x + 8 = 0$ are $x = -4$ and $x = -2$ .

Real-World Applications of Quadratic Equations

Quadratic equations have numerous real-world applications. Some examples include:

Projectile Motion: Quadratic equations are used to model the trajectory of projectiles, such as the path of a thrown ball or the trajectory of a rocket.
Optimization: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
Electrical Engineering: Quadratic equations are used to analyze electrical circuits and find the values of resistors and capacitors.
Computer Science: Quadratic equations are used in computer graphics and game development to create realistic simulations and animations.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have explored the solutions to the quadratic equation $x^2 = 16x - 65$ using the quadratic formula and completing the square. We have also discussed the importance of quadratic equations in real-world applications and provided a step-by-step guide on how to solve them. Whether you are a student or a professional, understanding quadratic equations is essential for success in mathematics and beyond.

Final Answer

The final answer to the quadratic equation $x^2 = 16x - 65$ is:

$x = 8 \pm i$

This is the solution to the equation, and it can be verified using the quadratic formula or completing the square.

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Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task. In this article, we will provide a Q&A guide to help you understand quadratic equations and solve them with ease.

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 is typically written in the 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: This involves expressing the quadratic equation as a product of two binomials.
Completing the Square: This involves rewriting the quadratic equation in a form that allows us to easily identify the solutions.
Quadratic Formula: This involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions.

Q: What is the Quadratic Formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

Q: How Do I Use the Quadratic Formula?

A: To use the quadratic formula, you need to:

Identify the coefficients $a$ , $b$ , and $c$ of the quadratic equation.
Plug these values into the quadratic formula.
Simplify the expression to find the solutions.

Q: What is Completing the Square?

A: Completing the square is a method for solving quadratic equations by rewriting them in a form that allows us to easily identify the solutions. It involves adding and subtracting a constant term to the equation.

Q: How Do I Complete the Square?

A: To complete the square, you need to:

Identify the quadratic equation.
Add and subtract a constant term to the equation.
Rewrite the equation in a form that allows us to easily identify the solutions.

Q: What are the Real-World Applications of Quadratic Equations?

A: Quadratic equations have numerous real-world applications, including:

Projectile Motion: Quadratic equations are used to model the trajectory of projectiles, such as the path of a thrown ball or the trajectory of a rocket.
Optimization: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
Electrical Engineering: Quadratic equations are used to analyze electrical circuits and find the values of resistors and capacitors.
Computer Science: Quadratic equations are used in computer graphics and game development to create realistic simulations and animations.

Q: How Do I Choose the Right Method for Solving a Quadratic Equation?

A: To choose the right method for solving a quadratic equation, you need to:

Identify the coefficients $a$ , $b$ , and $c$ of the quadratic equation.
Determine if the equation can be factored or if it requires the quadratic formula or completing the square.
Choose the method that is most suitable for the equation.

Q: What are the Common Mistakes to Avoid When Solving Quadratic Equations?

A: Some common mistakes to avoid when solving quadratic equations include:

Not identifying the coefficients $a$ , $b$ , and $c$ correctly.
Not using the correct method for solving the equation.
Not simplifying the expression correctly.
Not checking the solutions for validity.

Q: How Do I Check the Solutions for Validity?

A: To check the solutions for validity, you need to:

Plug the solutions back into the original equation.
Check if the equation is satisfied.
If the equation is not satisfied, the solution is invalid.

Q: What are the Benefits of Understanding Quadratic Equations?

A: Understanding quadratic equations has numerous benefits, including:

Improved problem-solving skills.
Enhanced critical thinking skills.
Increased confidence in mathematics.
Better preparation for advanced mathematics courses.

Q: How Can I Practice Solving Quadratic Equations?

A: To practice solving quadratic equations, you can:

Use online resources, such as quadratic equation solvers or practice problems.
Work with a tutor or mentor.
Join a study group or math club.
Practice solving quadratic equations regularly.

Q: What are the Next Steps After Solving Quadratic Equations?

A: After solving quadratic equations, you can:

Move on to more advanced topics, such as polynomial equations or systems of equations.
Apply quadratic equations to real-world problems, such as projectile motion or optimization.
Develop your problem-solving skills by working on more challenging problems.
Explore the connections between quadratic equations and other areas of mathematics, such as algebra or geometry.