Solve The Equation: 5 X + 39 = X + 9 \sqrt{5x + 39} = X + 9 5 X + 39 ​ = X + 9

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Introduction

In this article, we will delve into the world of algebra and explore the solution to a quadratic equation involving a square root. The equation $\sqrt{5x + 39} = x + 9$ may seem daunting at first, but with the right approach, we can break it down and find the value of $x$. This equation is a great example of how algebraic manipulations can be used to solve complex equations.

Understanding the Equation

The given equation is $\sqrt{5x + 39} = x + 9$. To solve this equation, we need to isolate the variable $x$. The first step is to get rid of the square root by squaring both sides of the equation. This will allow us to simplify the equation and make it easier to solve.

Squaring Both Sides

When we square both sides of the equation, we get:

$$\left(\sqrt{5x + 39}\right)^2 = (x + 9)^2$$

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

$$5x + 39 = (x + 9)^2$$

Expanding the Right-Hand Side

To expand the right-hand side of the equation, we need to use the formula $(a + b)^2 = a^2 + 2ab + b^2$. In this case, $a = x$ and $b = 9$. Therefore, we can expand the right-hand side as follows:

$$(x + 9)^2 = x^2 + 2 \cdot x \cdot 9 + 9^2$$

Simplifying the expression, we get:

$$(x + 9)^2 = x^2 + 18x + 81$$

Substituting the Expanded Expression

Now that we have expanded the right-hand side of the equation, we can substitute it back into the original equation:

$$5x + 39 = x^2 + 18x + 81$$

Rearranging the Equation

To make it easier to solve the equation, we can rearrange it by moving all the terms to one side of the equation:

$$x^2 + 18x + 81 - 5x - 39 = 0$$

Simplifying the expression, we get:

$$x^2 + 13x + 42 = 0$$

Factoring the Quadratic Equation

The quadratic equation $x^2 + 13x + 42 = 0$ can be factored as follows:

$$(x + 6)(x + 7) = 0$$

Solving for $x$

To solve for $x$, we need to set each factor equal to zero and solve for $x$:

$$x + 6 = 0 \Rightarrow x = -6$$

$$x + 7 = 0 \Rightarrow x = -7$$

Conclusion

In this article, we have solved the equation $\sqrt{5x + 39} = x + 9$ using algebraic manipulations. We started by squaring both sides of the equation, expanded the right-hand side, and rearranged the equation to make it easier to solve. Finally, we factored the quadratic equation and solved for $x$. The solutions to the equation are $x = -6$ and $x = -7$.

Final Answer

The final answer is $\boxed{-6}$ and $\boxed{-7}$.

Additional Tips and Tricks

• When solving equations involving square roots, it's often helpful to square both sides of the equation to get rid of the square root.
• When expanding expressions, use the formula $(a + b)^2 = a^2 + 2ab + b^2$ to simplify the expression.
• When rearranging equations, move all the terms to one side of the equation to make it easier to solve.
• When factoring quadratic equations, look for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Common Mistakes to Avoid

• When squaring both sides of the equation, make sure to square both sides of the equation, not just one side.
• When expanding expressions, make sure to use the correct formula and simplify the expression correctly.
• When rearranging equations, make sure to move all the terms to one side of the equation to avoid introducing extraneous solutions.
• When factoring quadratic equations, make sure to look for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Real-World Applications

Solving equations involving square roots has many real-world applications, including:

• Physics: When solving problems involving motion, it's often necessary to use equations involving square roots.
• Engineering: When designing structures, it's often necessary to use equations involving square roots to ensure that the structure can withstand various loads.
• Computer Science: When solving problems involving algorithms, it's often necessary to use equations involving square roots to optimize the algorithm.

Conclusion

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Introduction

In our previous article, we solved the equation $\sqrt{5x + 39} = x + 9$ using algebraic manipulations. In this article, we will answer some of the most frequently asked questions about solving this equation.

Q: What is the first step in solving the equation $\sqrt{5x + 39} = x + 9$?

A: The first step in solving the equation $\sqrt{5x + 39} = x + 9$ is to square both sides of the equation. This will allow us to get rid of the square root and simplify the equation.

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

A: We need to square both sides of the equation because the square root is a non-linear operation. By squaring both sides of the equation, we can eliminate the square root and make it easier to solve the equation.

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

A: After squaring both sides of the equation, we need to expand the right-hand side of the equation. This will allow us to simplify the equation and make it easier to solve.

Q: How do we expand the right-hand side of the equation?

A: We expand the right-hand side of the equation by using the formula $(a + b)^2 = a^2 + 2ab + b^2$. In this case, $a = x$ and $b = 9$. Therefore, we can expand the right-hand side as follows:

$$(x + 9)^2 = x^2 + 2 \cdot x \cdot 9 + 9^2$$

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

A: After expanding the right-hand side of the equation, we need to simplify the equation by combining like terms. This will allow us to make the equation easier to solve.

Q: How do we simplify the equation?

A: We simplify the equation by combining like terms. In this case, we can combine the terms $x^2$ and $18x$ to get $x^2 + 18x$. We can also combine the terms $81$ and $39$ to get $120$.

Q: What is the final step in solving the equation $\sqrt{5x + 39} = x + 9$?

A: The final step in solving the equation $\sqrt{5x + 39} = x + 9$ is to factor the quadratic equation. This will allow us to find the values of $x$ that satisfy the equation.

Q: How do we factor the quadratic equation?

A: We factor the quadratic equation by looking for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term. In this case, we can factor the quadratic equation as follows:

$$(x + 6)(x + 7) = 0$$

Q: What are the solutions to the equation $\sqrt{5x + 39} = x + 9$?

A: The solutions to the equation $\sqrt{5x + 39} = x + 9$ are $x = -6$ and $x = -7$.

Q: Why do we need to check the solutions to the equation?

A: We need to check the solutions to the equation because we squared both sides of the equation when we solved it. This means that we may have introduced extraneous solutions. By checking the solutions, we can make sure that they are valid.

Q: How do we check the solutions to the equation?

A: We check the solutions to the equation by plugging them back into the original equation. If the solution satisfies the original equation, then it is a valid solution.

Conclusion

In this article, we answered some of the most frequently asked questions about solving the equation $\sqrt{5x + 39} = x + 9$. We covered the steps involved in solving the equation, including squaring both sides of the equation, expanding the right-hand side, simplifying the equation, and factoring the quadratic equation. We also discussed how to check the solutions to the equation to make sure that they are valid.