Find The Solution(s) To $(x-3)^2=49$. Check All That Apply.A. $x=10$ B. $x=7$ C. $x=-10$ D. $x=-4$ E. $x=-7$

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 the quadratic equation $(x-3)^2=49$ and explore the different solutions that satisfy this equation.

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, the quadratic formula, and completing the square.

The Given Equation

The given equation is $(x-3)^2=49$. To solve this equation, we need to isolate the variable $x$. The first step is to expand the squared term using the formula $(a-b)^2 = a^2 - 2ab + b^2$. In this case, $a = x$ and $b = 3$, so we have:

$$(x-3)^2 = x^2 - 2 \cdot x \cdot 3 + 3^2$$

Simplifying the equation, we get:

$$x^2 - 6x + 9 = 49$$

Rearranging the Equation

To solve for $x$, we need to isolate the variable on one side of the equation. We can do this by subtracting 49 from both sides of the equation:

$$x^2 - 6x + 9 - 49 = 0$$

Simplifying the equation, we get:

$$x^2 - 6x - 40 = 0$$

Factoring the Equation

Now that we have a quadratic equation in the form $ax^2 + bx + c = 0$, we can try to factor it. Factoring a quadratic equation involves finding two binomials whose product is equal to the original equation. In this case, we can factor the equation as:

$$(x - 10)(x + 4) = 0$$

Solving for $x$

To solve for $x$, we need to set each factor equal to zero and solve for $x$. Setting the first factor equal to zero, we get:

$$x - 10 = 0$$

Solving for $x$, we get:

$$x = 10$$

Setting the second factor equal to zero, we get:

$$x + 4 = 0$$

Solving for $x$, we get:

$$x = -4$$

Checking the Solutions

Now that we have found two solutions, $x = 10$ and $x = -4$, we need to check if they satisfy the original equation. Plugging $x = 10$ into the original equation, we get:

$$(10-3)^2 = 49$$

Simplifying the equation, we get:

$$7^2 = 49$$

This is true, so $x = 10$ is a solution to the equation.

Plugging $x = -4$ into the original equation, we get:

$$( -4 - 3)^2 = 49$$

Simplifying the equation, we get:

$$(-7)^2 = 49$$

This is also true, so $x = -4$ is a solution to the equation.

Conclusion

In this article, we solved the quadratic equation $(x-3)^2=49$ and found two solutions: $x = 10$ and $x = -4$. We also checked these solutions to ensure that they satisfy the original equation. This example illustrates the importance of carefully solving quadratic equations and checking the solutions to ensure that they are correct.

Final Answer

The final answer is: $\boxed{A, D}$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations and provide detailed answers to help you better understand this topic.

Q1: What is a quadratic equation?

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.

Q2: How do I solve a quadratic equation?

There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and completing the square. The method you choose will depend on the specific equation and your personal preference.

Q3: What is the quadratic formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. The formula is:

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

Q4: How do I use the quadratic formula?

To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, plug these values into the formula and simplify to find the solutions.

Q5: What is the difference between a quadratic equation and a linear equation?

A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared term, while a linear equation does not.

Q6: Can I solve a quadratic equation by graphing?

Yes, you can solve a quadratic equation by graphing. By graphing the quadratic function, you can find the x-intercepts, which represent the solutions to the equation.

Q7: What is the significance of the discriminant in a quadratic equation?

The discriminant is the expression under the square root in 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.

Q8: Can I solve a quadratic equation with complex solutions?

Yes, you can solve a quadratic equation with complex solutions. In this case, the solutions will be complex numbers, which are numbers that have both real and imaginary parts.

Q9: How do I check my solutions to a quadratic equation?

To check your solutions, plug the values back into the original equation and simplify. If the equation is true, then the solution is correct.

Q10: What are some common mistakes to avoid when solving quadratic equations?

Some common mistakes to avoid when solving quadratic equations include:

• Not simplifying the equation properly
• Not checking the solutions
• Not using the correct method for solving the equation
• Not considering complex solutions

Conclusion

In this article, we have addressed some of the most frequently asked questions about quadratic equations and provided detailed answers to help you better understand this topic. By following the tips and techniques outlined in this article, you will be able to solve quadratic equations with confidence and accuracy.

Final Answer

The final answer is: There is no final answer, as this article is a Q&A and does not have a specific numerical solution.