Solve The Equation: 3 X + 13 = X + 3 \sqrt{3x+13}=x+3 3 X + 13 ​ = X + 3

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Introduction

Solving equations involving square roots can be a challenging task, especially when the equation is not straightforward. In this article, we will focus on solving the equation $\sqrt{3x+13}=x+3$. This equation involves a square root, and our goal is to isolate the variable $x$ and find its value. We will use algebraic techniques to solve this equation, and we will also provide a step-by-step solution to help you understand the process.

Understanding the Equation

The given equation is $\sqrt{3x+13}=x+3$. This equation involves a square root, which means that the expression inside the square root must be non-negative. In other words, $3x+13 \geq 0$. We will assume that this condition is satisfied, and we will proceed with solving the equation.

Solving the Equation

To solve the equation, we will start by isolating the square root expression. We can do this by squaring both sides of the equation. This will eliminate the square root and allow us to solve for $x$.

Step 1: Square Both Sides

We will start by squaring both sides of the equation:

$$\left(\sqrt{3x+13}\right)^2 = (x+3)^2$$

Using the property of exponents, we can simplify the left-hand side of the equation:

$$3x+13 = (x+3)^2$$

Step 2: Expand the Right-Hand Side

We will expand the right-hand side of the equation using the binomial theorem:

$$(x+3)^2 = x^2 + 6x + 9$$

Substituting this expression into the equation, we get:

$$3x+13 = x^2 + 6x + 9$$

Step 3: Rearrange the Equation

We will rearrange the equation to get a quadratic equation in $x$:

$$x^2 + 3x - 4 = 0$$

Step 4: Solve the Quadratic Equation

We will solve the quadratic equation using the quadratic formula:

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

In this case, $a=1$, $b=3$, and $c=-4$. Substituting these values into the formula, we get:

$$x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-4)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{-3 \pm \sqrt{25}}{2}$$

$$x = \frac{-3 \pm 5}{2}$$

Step 5: Find the Solutions

We will find the two possible solutions for $x$:

$$x_1 = \frac{-3 + 5}{2} = 1$$

$$x_2 = \frac{-3 - 5}{2} = -4$$

Checking the Solutions

We will check the solutions to make sure they satisfy the original equation. We will substitute each solution into the original equation and check if it is true.

Checking $x_1=1$

We will substitute $x_1=1$ into the original equation:

$$\sqrt{3(1)+13} = 1+3$$

Simplifying the expression, we get:

$$\sqrt{16} = 4$$

This is true, so $x_1=1$ is a valid solution.

Checking $x_2=-4$

We will substitute $x_2=-4$ into the original equation:

$$\sqrt{3(-4)+13} = -4+3$$

Simplifying the expression, we get:

$$\sqrt{1} = -1$$

This is not true, so $x_2=-4$ is not a valid solution.

Conclusion

We have solved the equation $\sqrt{3x+13}=x+3$ using algebraic techniques. We have found one valid solution, $x_1=1$, and we have checked that it satisfies the original equation. We have also found that $x_2=-4$ is not a valid solution. We hope that this article has helped you understand how to solve equations involving square roots.

Final Answer

The final answer is $\boxed{1}$.

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Introduction

In our previous article, we solved the equation $\sqrt{3x+13}=x+3$ using algebraic techniques. We found one valid solution, $x_1=1$, and we checked that it satisfies the original equation. In this article, we will provide a Q&A section to help you understand the solution and to answer any questions you may have.

Q&A

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

A: The first step in solving the equation is to isolate the square root expression. We can do this by squaring both sides of the equation.

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

A: We square both sides of the equation to eliminate the square root and to make it easier to solve for $x$.

Q: What is the next step after squaring both sides of the equation?

A: After squaring both sides of the equation, we expand the right-hand side of the equation using the binomial theorem.

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical formula that allows us to expand expressions of the form $(a+b)^n$.

Q: How do we use the binomial theorem to expand the right-hand side of the equation?

A: We use the binomial theorem to expand the right-hand side of the equation by substituting $a=x$ and $b=3$ into the formula.

Q: What is the resulting equation after expanding the right-hand side of the equation?

A: The resulting equation after expanding the right-hand side of the equation is $x^2 + 3x - 4 = 0$.

Q: How do we solve the quadratic equation?

A: We solve the quadratic equation using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that allows us to solve quadratic equations of the form $ax^2 + bx + c = 0$.

Q: How do we use the quadratic formula to solve the quadratic equation?

A: We use the quadratic formula to solve the quadratic equation by substituting $a=1$, $b=3$, and $c=-4$ into the formula.

Q: What are the two possible solutions for $x$?

A: The two possible solutions for $x$ are $x_1 = \frac{-3 + 5}{2} = 1$ and $x_2 = \frac{-3 - 5}{2} = -4$.

Q: How do we check the solutions to make sure they satisfy the original equation?

A: We check the solutions by substituting each solution into the original equation and checking if it is true.

Q: Which solution satisfies the original equation?

A: Only $x_1=1$ satisfies the original equation.

Conclusion

We hope that this Q&A article has helped you understand the solution to the equation $\sqrt{3x+13}=x+3$. If you have any further questions or need clarification on any of the steps, please don't hesitate to ask.

Final Answer

The final answer is $\boxed{1}$.