What Is The Solution Of $4+\sqrt{5x+66}=x+10$?A. $x = -10$ B. $x = 3$ C. $x = -10$ Or $x = 3$ D. No Solution

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Introduction

In this article, we will explore the solution of the given equation, which involves a square root. The equation is $4+\sqrt{5x+66}=x+10$. Our goal is to find the value of xx that satisfies this equation. We will use algebraic techniques to isolate the variable xx and find its possible values.

Understanding the Equation

The given equation is a quadratic equation in disguise. It involves a square root, which can be eliminated by squaring both sides of the equation. However, before we do that, let's analyze the equation and understand its structure.

The equation is $4+\sqrt{5x+66}=x+10$. We can see that the left-hand side of the equation involves a square root, while the right-hand side is a linear expression in xx. Our goal is to eliminate the square root and find the value of xx that satisfies the equation.

Isolating the Square Root

To eliminate the square root, we can isolate it on one side of the equation. We can do this by subtracting 44 from both sides of the equation, which gives us:

5x+66=x+6\sqrt{5x+66}=x+6

Squaring Both Sides

Now that we have isolated the square root, we can eliminate it by squaring both sides of the equation. This will give us a quadratic equation in xx, which we can solve to find the value of xx.

(5x+66)2=(x+6)2\left(\sqrt{5x+66}\right)^2=\left(x+6\right)^2

Expanding the right-hand side of the equation, we get:

5x+66=x2+12x+365x+66=x^2+12x+36

Simplifying the Equation

Now that we have expanded the right-hand side of the equation, we can simplify it by combining like terms. This will give us a quadratic equation in xx, which we can solve to find the value of xx.

5x+66=x2+12x+365x+66=x^2+12x+36

Subtracting 5x5x from both sides of the equation, we get:

66=x2+7x+3666=x^2+7x+36

Subtracting 3636 from both sides of the equation, we get:

30=x2+7x30=x^2+7x

Solving the Quadratic Equation

Now that we have simplified the equation, we can solve it to find the value of xx. We can do this by factoring the quadratic expression or by using the quadratic formula.

Let's try factoring the quadratic expression:

x2+7x−30=0x^2+7x-30=0

Factoring the left-hand side of the equation, we get:

(x+10)(x−3)=0(x+10)(x-3)=0

Finding the Solutions

Now that we have factored the quadratic expression, we can find the solutions to the equation. We can do this by setting each factor equal to zero and solving for xx.

Setting the first factor equal to zero, we get:

x+10=0x+10=0

Solving for xx, we get:

x=−10x=-10

Setting the second factor equal to zero, we get:

x−3=0x-3=0

Solving for xx, we get:

x=3x=3

Conclusion

In this article, we have explored the solution of the given equation, which involves a square root. We have used algebraic techniques to isolate the variable xx and find its possible values. We have found that the equation has two solutions: x=−10x=-10 and x=3x=3. Therefore, the correct answer is:

C. x=−10x = -10 or x=3x = 3

Discussion

The given equation is a quadratic equation in disguise. It involves a square root, which can be eliminated by squaring both sides of the equation. However, before we do that, we need to analyze the equation and understand its structure. The equation is $4+\sqrt{5x+66}=x+10$. We can see that the left-hand side of the equation involves a square root, while the right-hand side is a linear expression in xx. Our goal is to eliminate the square root and find the value of xx that satisfies the equation.

The equation can be solved by isolating the square root and then squaring both sides of the equation. This will give us a quadratic equation in xx, which we can solve to find the value of xx. The quadratic equation can be factored or solved using the quadratic formula.

In this article, we have used algebraic techniques to isolate the variable xx and find its possible values. We have found that the equation has two solutions: x=−10x=-10 and x=3x=3. Therefore, the correct answer is:

C. x=−10x = -10 or x=3x = 3

Final Answer

The final answer is C. x=−10x = -10 or x=3x = 3.

Introduction

In our previous article, we explored the solution of the given equation, which involves a square root. We used algebraic techniques to isolate the variable xx and find its possible values. In this article, we will answer some frequently asked questions about the solution of the equation.

Q: What is the equation $4+\sqrt{5x+66}=x+10$?

A: The equation is a quadratic equation in disguise. It involves a square root, which can be eliminated by squaring both sides of the equation.

Q: How do I solve the equation $4+\sqrt{5x+66}=x+10$?

A: To solve the equation, you need to isolate the square root and then square both sides of the equation. This will give you a quadratic equation in xx, which you can solve to find the value of xx.

Q: What are the steps to solve the equation $4+\sqrt{5x+66}=x+10$?

A: The steps to solve the equation are:

  1. Isolate the square root by subtracting 44 from both sides of the equation.
  2. Square both sides of the equation to eliminate the square root.
  3. Simplify the equation by combining like terms.
  4. Solve the quadratic equation to find the value of xx.

Q: What are the solutions to the equation $4+\sqrt{5x+66}=x+10$?

A: The solutions to the equation are x=−10x=-10 and x=3x=3.

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

A: You need to square both sides of the equation to eliminate the square root. This will give you a quadratic equation in xx, which you can solve to find the value of xx.

Q: Can I use the quadratic formula to solve the equation?

A: Yes, you can use the quadratic formula to solve the equation. The quadratic formula is:

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

You can plug in the values of aa, bb, and cc into the quadratic formula to find the value of xx.

Q: What is the final answer to the equation $4+\sqrt{5x+66}=x+10$?

A: The final answer to the equation is x=−10x=-10 or x=3x=3.

Conclusion

In this article, we have answered some frequently asked questions about the solution of the equation $4+\sqrt{5x+66}=x+10$. We have provided step-by-step instructions on how to solve the equation and have discussed the solutions to the equation. We hope that this article has been helpful in understanding the solution of the equation.

Final Answer

The final answer is C. x=−10x = -10 or x=3x = 3.