Solve The Equation:$\sqrt{x+6} + 1 = X + 10$

Introduction

In mathematics, equations are a fundamental concept that help us understand and describe various phenomena. Solving equations is a crucial skill that is essential in many areas of mathematics, science, and engineering. In this article, we will focus on solving a specific equation, $\sqrt{x+6} + 1 = x + 10$, and provide a step-by-step guide on how to approach it.

Understanding the Equation

The given equation is $\sqrt{x+6} + 1 = x + 10$. This equation involves a square root, which can be challenging to solve. However, with a systematic approach, we can break down the equation and solve for the variable $x$.

Step 1: Isolate the Square Root

The first step is to isolate the square root term on one side of the equation. We can do this by subtracting 1 from both sides of the equation:

$$\sqrt{x+6} = x + 10 - 1$$

This simplifies to:

$$\sqrt{x+6} = x + 9$$

Step 2: Square Both Sides

To eliminate the square root, we can square both sides of the equation:

$$(\sqrt{x+6})^2 = (x + 9)^2$$

This simplifies to:

$$x + 6 = x^2 + 18x + 81$$

Step 3: Rearrange the Equation

Now, we can rearrange the equation to get all the terms on one side:

$$x^2 + 18x + 81 - x - 6 = 0$$

This simplifies to:

$$x^2 + 17x + 75 = 0$$

Step 4: Solve the Quadratic Equation

The resulting equation is a quadratic equation in the form $ax^2 + bx + c = 0$. We can solve this equation using various methods, such as factoring, completing the square, or using the quadratic formula.

Method 1: Factoring

Let's try to factor the quadratic equation:

$$x^2 + 17x + 75 = (x + 15)(x + 5) = 0$$

This gives us two possible solutions:

$$x + 15 = 0 \quad \text{or} \quad x + 5 = 0$$

Solving for $x$, we get:

$$x = -15 \quad \text{or} \quad x = -5$$

Method 2: Quadratic Formula

Alternatively, we can use the quadratic formula to solve the equation:

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

In this case, $a = 1$, $b = 17$, and $c = 75$. Plugging these values into the formula, we get:

$$x = \frac{-17 \pm \sqrt{17^2 - 4(1)(75)}}{2(1)}$$

Simplifying, we get:

$$x = \frac{-17 \pm \sqrt{289 - 300}}{2}$$

This simplifies to:

$$x = \frac{-17 \pm \sqrt{-11}}{2}$$

Since the square root of a negative number is not a real number, this solution is not valid.

Conclusion

In this article, we solved the equation $\sqrt{x+6} + 1 = x + 10$ using a step-by-step approach. We isolated the square root term, squared both sides, and rearranged the equation to get a quadratic equation. We then solved the quadratic equation using factoring and the quadratic formula. The solutions to the equation are $x = -15$ and $x = -5$.

Final Answer

$$$$

Introduction

In our previous article, we solved the equation $\sqrt{x+6} + 1 = x + 10$ using a step-by-step approach. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

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

A: The first step is to isolate the square root term on one side of the equation. We can do this by subtracting 1 from both sides of the equation.

Q: Why do we need to square both sides of the equation?

A: We need to square both sides of the equation to eliminate the square root. This is because the square root of a number is equal to the number raised to the power of 1/2. By squaring both sides, we can get rid of the square root and solve for the variable.

Q: What is the quadratic formula, and how do we use it to solve the equation?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

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

In this case, $a = 1$, $b = 17$, and $c = 75$. We plug these values into the formula and simplify to get the solutions.

Q: Why do we get two solutions for the equation?

A: We get two solutions for the equation because the quadratic formula gives us two possible values for the variable. In this case, the two solutions are $x = -15$ and $x = -5$.

Q: How do we know which solution is correct?

A: We know which solution is correct by plugging the values back into the original equation and checking if they satisfy the equation. In this case, both $x = -15$ and $x = -5$ satisfy the equation.

Q: What if the quadratic formula gives us a complex solution?

A: If the quadratic formula gives us a complex solution, it means that the equation has no real solutions. In this case, we can say that the equation has no real solutions, and we can stop there.

Q: Can we use other methods to solve the equation?

A: Yes, we can use other methods to solve the equation, such as factoring or completing the square. However, the quadratic formula is a powerful tool that can be used to solve quadratic equations quickly and easily.

Q: How do we apply the solution to real-world problems?

A: The solution to the equation can be applied to real-world problems in various fields, such as physics, engineering, and economics. For example, the solution can be used to model population growth, chemical reactions, or financial transactions.

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the equation $\sqrt{x+6} + 1 = x + 10$. We answered questions about the first step in solving the equation, the quadratic formula, and how to apply the solution to real-world problems.

Final Answer

The final answer is $\boxed{-15}$ and $\boxed{-5}$.