Select The Correct Answer.What Are The Solutions Of This Quadratic Equation? X 2 + 4 = 8 X + 5 X^2 + 4 = 8x + 5 X 2 + 4 = 8 X + 5 A. X = 8 ± 34 X = 8 \pm \sqrt{34} X = 8 ± 34 ​ B. X = 8 ± 2 17 X = 8 \pm 2\sqrt{17} X = 8 ± 2 17 ​ C. X = 4 ± 7 X = 4 \pm \sqrt{7} X = 4 ± 7 ​ D. X = 4 ± 17 X = 4 \pm \sqrt{17} X = 4 ± 17 ​

Introduction

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 focus on solving a specific quadratic equation, $x^2 + 4 = 8x + 5$, and explore the different solutions that can be obtained.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, 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 states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

This formula can be used to solve any quadratic equation, regardless of whether it can be factored or not.

Solving the Given Quadratic Equation

Now, let's apply the quadratic formula to the given equation, $x^2 + 4 = 8x + 5$. First, we need to rewrite the equation in the standard form, $ax^2 + bx + c = 0$. We can do this by subtracting $8x$ from both sides and adding $5$ to both sides:

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

Now, we can identify the values of $a$, $b$, and $c$: $a = 1$, $b = -8$, and $c = 9$.

Applying the Quadratic Formula

Substituting these values into the quadratic formula, we get:

$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(9)}}{2(1)}$

Simplifying the expression, we get:

$x = \frac{8 \pm \sqrt{64 - 36}}{2}$

$x = \frac{8 \pm \sqrt{28}}{2}$

$x = \frac{8 \pm 2\sqrt{7}}{2}$

$x = 4 \pm \sqrt{7}$

Conclusion

In this article, we have solved the quadratic equation $x^2 + 4 = 8x + 5$ using the quadratic formula. We have obtained two possible solutions: $x = 4 + \sqrt{7}$ and $x = 4 - \sqrt{7}$. These solutions are in the form of $x = a \pm \sqrt{b}$, where $a$ and $b$ are constants.

Answer

The correct answer is:

• C. $x = 4 \pm \sqrt{7}$

This solution is obtained by applying the quadratic formula to the given equation and simplifying the expression.

Discussion

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 focused on solving a specific quadratic equation using the quadratic formula. We have obtained two possible solutions and have discussed the different methods for solving quadratic equations.

Additional Resources

For more information on quadratic equations and the quadratic formula, please refer to the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Formula
• Wolfram Alpha: Quadratic Equation Solver

Final Thoughts

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we focused on solving a specific quadratic equation using the quadratic formula. In this article, we will provide a Q&A guide to help you better understand quadratic equations and the quadratic formula.

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 (in this case, $x$) 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 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 apply the quadratic formula?

A: To apply the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the given equation. Then, substitute these values into the quadratic formula and simplify the expression.

Q: What are the different methods for solving quadratic equations?

A: There are several methods for solving quadratic equations, including:

• Factoring: This method involves expressing the quadratic equation as a product of two binomials.
• Completing the square: This method involves rewriting the quadratic equation in the form $(x + p)^2 + q = 0$.
• Quadratic formula: This method involves using the quadratic formula to find the solutions.

Q: What are the advantages and disadvantages of the quadratic formula?

A: The advantages of the quadratic formula include:

• It can be used to solve any quadratic equation, regardless of whether it can be factored or not.
• It provides two possible solutions for the equation.

The disadvantages of the quadratic formula include:

• It can be complex to apply, especially for equations with large coefficients.
• It may not provide a clear understanding of the underlying mathematics.

Q: How do I choose the correct method for solving a quadratic equation?

A: To choose the correct method for solving a quadratic equation, you need to consider the following factors:

• The complexity of the equation: If the equation is simple, factoring or completing the square may be sufficient. If the equation is complex, the quadratic formula may be more suitable.
• The desired level of accuracy: If you need a high level of accuracy, the quadratic formula may be more suitable.
• The level of understanding: If you want to gain a deeper understanding of the underlying mathematics, completing the square or factoring may be more suitable.

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

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

• Not identifying the values of $a$, $b$, and $c$ correctly.
• Not simplifying the expression correctly.
• Not considering the possibility of complex solutions.

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 provided a Q&A guide to help you better understand quadratic equations and the quadratic formula. We hope that this article has provided valuable insights and information on quadratic equations and the quadratic formula.

Additional Resources

For more information on quadratic equations and the quadratic formula, please refer to the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Formula
• Wolfram Alpha: Quadratic Equation Solver

Final Thoughts

Solving quadratic equations is a crucial skill for students and professionals alike. In this article, we have provided a Q&A guide to help you better understand quadratic equations and the quadratic formula. We hope that this article has provided valuable insights and information on quadratic equations and the quadratic formula.