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. Enter The Correct Answer In The Box.

=====================================================

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

The Quadratic Formula

---

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

In our case, we have $a = 1$, $b = -16$, and $c = 65$. Substituting 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:

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

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

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

$x = 8 \pm i$

Solutions of the Quadratic Equation

---

The solutions of the quadratic equation $x^2 = 16x - 65$ are $8 + i$ and $8 - i$. These solutions are complex numbers, which means they have both real and imaginary parts.

Conclusion

---

In this article, we have solved a quadratic equation using the quadratic formula. We have shown that the solutions of the equation $x^2 = 16x - 65$ are $8 + i$ and $8 - i$. These solutions are complex numbers, which means they have both real and imaginary parts.

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. 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.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

Q: What are complex numbers?

A: Complex numbers are numbers that have both real and imaginary parts. They are used to solve equations that have no real solutions.

References

---

• [1] "Quadratic Equations" by Math Open Reference
• [2] "The Quadratic Formula" by Purplemath
• [3] "Complex Numbers" by Khan Academy

Further Reading

---

• "Quadratic Equations and Functions" by Math Is Fun
• "The Quadratic Formula and Complex Numbers" by IXL
• "Quadratic Equations and Inequalities" by CK-12 Foundation
  

=========================

Introduction

---

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we solved a quadratic equation using the quadratic formula and found the solutions to be complex numbers. In this article, we will answer some frequently asked questions about quadratic equations and provide additional information to help you better understand this topic.

Q&A

---

Q: What is the difference between a quadratic equation and a linear equation?

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable is one. The general form of a linear equation is $ax + b = 0$, where $a$ and $b$ are constants, and $x$ is the variable. A quadratic equation, on the other hand, 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.

Q: How do I determine if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to substitute the values of $a$, $b$, and $c$ into the formula and simplify the expression. The solutions to the equation will be given by the two values of $x$ that satisfy the equation.

Q: What are some common mistakes to avoid when solving quadratic equations?

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

• Not simplifying the expression under the square root in the quadratic formula
• Not checking if the discriminant is positive, zero, or negative before solving the equation
• Not using the correct values of $a$, $b$, and $c$ in the quadratic formula
• Not simplifying the solutions to the equation

Tips and Tricks

---

• Always simplify the expression under the square root in the quadratic formula before solving the equation.
• Check if the discriminant is positive, zero, or negative before solving the equation.
• Use the correct values of $a$, $b$, and $c$ in the quadratic formula.
• Simplify the solutions to the equation before presenting them.

Conclusion

---

In this article, we have answered some frequently asked questions about quadratic equations and provided additional information to help you better understand this topic. We have also discussed some common mistakes to avoid when solving quadratic equations and provided some tips and tricks to help you solve these equations more efficiently.

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. 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.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

Q: What are complex numbers?

A: Complex numbers are numbers that have both real and imaginary parts. They are used to solve equations that have no real solutions.

References

---

• [1] "Quadratic Equations" by Math Open Reference
• [2] "The Quadratic Formula" by Purplemath
• [3] "Complex Numbers" by Khan Academy

Further Reading

---

• "Quadratic Equations and Functions" by Math Is Fun
• "The Quadratic Formula and Complex Numbers" by IXL
• "Quadratic Equations and Inequalities" by CK-12 Foundation