Quadratic equation with complex roots

Oct 13, 2025 by windailys.com 73 views

Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the different methods and techniques used to solve them. In this article, we will focus on solving the quadratic equation $$(x-8)^2+6=56$$, and we will explore the different solutions and their validity.

Understanding the Equation

The given equation is a quadratic equation in the form of $$(x-a)^2+b=c$$. To solve this equation, we need to isolate the variable $x$ and find its possible values. The first step is to simplify the equation by subtracting $6$ from both sides, which gives us $$(x-8)^2=50$$.

Expanding the Equation

To expand the equation, we need to multiply the squared term by itself, which gives us $$(x-8)(x-8)=50$$. This can be further simplified to $$(x-8)^2=50$$.

Taking the Square Root

To solve for $x$ , we need to take the square root of both sides of the equation. This gives us $x-8=\pm\sqrt{50}$ .

Simplifying the Square Root

The square root of $50$ can be simplified as $\sqrt{50}=\sqrt{25\cdot2}=\sqrt{25}\cdot\sqrt{2}=5\sqrt{2}$ . Therefore, we have $x-8=\pm5\sqrt{2}$ .

Adding 8 to Both Sides

To isolate $x$ , we need to add $8$ to both sides of the equation, which gives us $x=8\pm5\sqrt{2}$ .

Conclusion

In conclusion, the solutions to the quadratic equation $$(x-8)^2+6=56$$ are $x=8\pm5\sqrt{2}$ . These solutions are valid and can be verified by plugging them back into the original equation.

Comparison with Answer Choices

Let's compare our solutions with the answer choices provided:

A. $x=-2$ and $x=18$

B. $x=8 \pm 5 \sqrt{2}$

C. $x= \pm \sqrt{89}$

D. $x=3$ and $x=13$

Our solutions match with answer choice B, which is $x=8 \pm 5 \sqrt{2}$ .

Final Answer

The final answer to the quadratic equation $$(x-8)^2+6=56$$ is $x=8\pm5\sqrt{2}$ .

Frequently Asked Questions

Q: What is the first step in solving the quadratic equation $$(x-8)^2+6=56$$?

A: The first step is to simplify the equation by subtracting $6$ from both sides, which gives us $$(x-8)^2=50$$.

Q: How do we expand the equation $$(x-8)^2=50$$?

A: We multiply the squared term by itself, which gives us $$(x-8)(x-8)=50$$.

Q: What is the next step in solving the equation?

A: We take the square root of both sides of the equation, which gives us $x-8=\pm\sqrt{50}$ .

Q: How do we simplify the square root of $50$ ?

A: We simplify the square root of $50$ as $\sqrt{50}=\sqrt{25\cdot2}=\sqrt{25}\cdot\sqrt{2}=5\sqrt{2}$ .

Q: What are the final solutions to the quadratic equation?

A: The final solutions to the quadratic equation are $x=8\pm5\sqrt{2}$ .

Conclusion

In conclusion, solving the quadratic equation $$(x-8)^2+6=56$$ requires a step-by-step approach, starting from simplifying the equation to taking the square root and finally isolating the variable $x$ . The final solutions to the equation are $x=8\pm5\sqrt{2}$ , which match with answer choice B.

Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the different methods and techniques used to solve them. In this article, we will provide a comprehensive Q&A guide to help you understand the solutions to quadratic equations.

Q&A Guide

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 of $ax^2+bx+c=0$ , where $a$ , $b$ , and $c$ are constants.

Q: What are the different types of quadratic equations?

A: There are three main types of quadratic equations:

Monic quadratic equation: A monic quadratic equation is a quadratic equation of the form $x^2+bx+c=0$ , where the coefficient of the $x^2$ term is 1.
Non-monic quadratic equation: A non-monic quadratic equation is a quadratic equation of the form $ax^2+bx+c=0$ , where the coefficient of the $x^2$ term is not 1.
Quadratic equation with complex roots: A quadratic equation with complex roots is a quadratic equation that has complex roots, which means the discriminant is negative.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following methods:

Factoring: If the quadratic equation can be factored, you can solve it by finding the factors.
Quadratic formula: If the quadratic equation cannot be factored, you can use the quadratic formula to solve it.
Graphing: You can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve a quadratic equation of the form $ax^2+bx+c=0$ . It is given by:

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

Q: What is the discriminant?

A: The discriminant is a value that can be calculated from the coefficients of a quadratic equation. It is given by:

\Delta=b^2-4ac

If the discriminant is positive, the quadratic equation has two real roots. If the discriminant is zero, the quadratic equation has one real root. If the discriminant is negative, the quadratic equation has two complex roots.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you can use the discriminant. If the discriminant is:

Positive, the quadratic equation has two real solutions.
Zero, the quadratic equation has one real solution.
Negative, the quadratic equation has two complex solutions.

Q: What are the solutions to the quadratic equation $$(x-8)^2+6=56$$?

A: The solutions to the quadratic equation $$(x-8)^2+6=56$$ are $x=8\pm5\sqrt{2}$ .

Q: How do I verify the solutions to a quadratic equation?

A: To verify the solutions to a quadratic equation, you can plug the solutions back into the original equation and check if they are true.

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 solutions before verifying them.
Not checking if the solutions are real or complex.

Conclusion

In conclusion, solving quadratic equations requires a step-by-step approach, starting from simplifying the equation to using the quadratic formula and finally verifying the solutions. By following the Q&A guide provided in this article, you can gain a better understanding of the solutions to quadratic equations and avoid common mistakes.

Frequently Asked Questions

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you can use the discriminant.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve a quadratic equation of the form $ax^2+bx+c=0$ .

Q: How do I verify the solutions to a quadratic equation?

A: To verify the solutions to a quadratic equation, you can plug the solutions back into the original equation and check if they are true.

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 solutions before verifying them, and not checking if the solutions are real or complex.