Solve For X X X . ( 5 X + 6 ) ( 3 X − 6 ) = 0 (5x + 6)(3x - 6) = 0 ( 5 X + 6 ) ( 3 X − 6 ) = 0 A. X = − 6 X = -6 X = − 6 Or X = 6 X = 6 X = 6 B. X = 0 X = 0 X = 0 C. X = − 6 5 X = \frac{-6}{5} X = 5 − 6 ​ Or X = 2 X = 2 X = 2 D. X = 6 5 X = \frac{6}{5} X = 5 6 ​ Or X = − 2 X = -2 X = − 2

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Introduction

In algebra, solving for $x$ in an equation is a fundamental concept that helps us find the value of the variable. When we have a quadratic equation or a product of two binomials equal to zero, we can use various techniques to find the solutions. In this article, we will explore how to solve for $x$ in the equation $(5x + 6)(3x - 6) = 0$.

Understanding the Equation

The given equation is a product of two binomials, which can be expanded using the distributive property. However, since we are given that the product is equal to zero, we can use the zero-product property to find the solutions. The zero-product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero.

Applying the Zero-Product Property

Using the zero-product property, we can set each factor equal to zero and solve for $x$. This gives us two separate equations:

1. $5x + 6 = 0$
2. $3x - 6 = 0$

Solving the First Equation

To solve the first equation, we can isolate the variable $x$ by subtracting 6 from both sides and then dividing by 5.

$$5x + 6 = 0$$

Subtracting 6 from both sides:

$$5x = -6$$

Dividing by 5:

$$x = \frac{-6}{5}$$

Solving the Second Equation

To solve the second equation, we can isolate the variable $x$ by adding 6 to both sides and then dividing by 3.

$$3x - 6 = 0$$

Adding 6 to both sides:

$$3x = 6$$

Dividing by 3:

$$x = 2$$

Conclusion

In conclusion, we have solved the equation $(5x + 6)(3x - 6) = 0$ using the zero-product property. We found that the solutions are $x = \frac{-6}{5}$ and $x = 2$. These values satisfy the original equation, and we can verify this by plugging them back into the equation.

Final Answer

The final answer is:

C. $x = \frac{-6}{5}$ or $x = 2$

Discussion

This problem is a great example of how to apply the zero-product property to solve for $x$ in a quadratic equation. By setting each factor equal to zero and solving for $x$, we can find the solutions to the equation. This technique is useful in many areas of mathematics, including algebra and calculus.

Related Problems

If you are looking for more practice problems on solving for $x$ in quadratic equations, here are a few related problems:

• Solve for $x$: $(2x + 3)(x - 4) = 0$
• Solve for $x$: $(x + 2)(3x - 1) = 0$
• Solve for $x$: $(x - 1)(2x + 5) = 0$

These problems require the same techniques as the original problem, and they are a great way to practice solving for $x$ in quadratic equations.

Tips and Tricks

Here are a few tips and tricks to help you solve for $x$ in quadratic equations:

• Always use the zero-product property to find the solutions.
• Set each factor equal to zero and solve for $x$.
• Use algebraic techniques, such as adding, subtracting, multiplying, and dividing, to isolate the variable $x$.
• Verify your solutions by plugging them back into the original equation.

By following these tips and tricks, you can become proficient in solving for $x$ in quadratic equations and apply this technique to a wide range of mathematical problems.

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Introduction

In our previous article, we explored how to solve for $x$ in the equation $(5x + 6)(3x - 6) = 0$. We used the zero-product property to find the solutions, which were $x = \frac{-6}{5}$ and $x = 2$. In this article, we will answer some frequently asked questions about solving for $x$ in quadratic equations.

Q&A

Q: What is the zero-product property?

A: The zero-product property is a fundamental concept in algebra that states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero.

Q: How do I apply the zero-product property to solve for $x$?

A: To apply the zero-product property, you need to set each factor equal to zero and solve for $x$. This will give you the solutions to the equation.

Q: What if I have a quadratic equation in the form of $ax^2 + bx + c = 0$? How do I solve for $x$?

A: To solve a quadratic equation in the form of $ax^2 + bx + c = 0$, you can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula will give you the solutions to the equation.

Q: Can I use the zero-product property to solve a quadratic equation in the form of $ax^2 + bx + c = 0$?

A: No, the zero-product property only applies to equations in the form of $ab = 0$, where $a$ and $b$ are factors. You need to use the quadratic formula to solve a quadratic equation in the form of $ax^2 + bx + c = 0$.

Q: How do I verify my solutions?

A: To verify your solutions, you need to plug them back into the original equation and check if they satisfy the equation. If they do, then they are valid solutions.

Q: What if I have a quadratic equation with complex solutions? How do I solve for $x$?

A: If you have a quadratic equation with complex solutions, you can use the quadratic formula to find the solutions. The quadratic formula will give you the complex solutions in the form of $x = a \pm bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: Can I use a calculator to solve for $x$ in a quadratic equation?

A: Yes, you can use a calculator to solve for $x$ in a quadratic equation. Most calculators have a built-in quadratic formula function that you can use to find the solutions.

Conclusion

In conclusion, solving for $x$ in quadratic equations is a fundamental concept in algebra that requires the use of various techniques, including the zero-product property and the quadratic formula. By understanding these techniques and practicing them, you can become proficient in solving for $x$ in quadratic equations and apply this technique to a wide range of mathematical problems.

Final Answer

The final answer is:

C. $x = \frac{-6}{5}$ or $x = 2$

Discussion

This article provides a comprehensive overview of how to solve for $x$ in quadratic equations, including the use of the zero-product property and the quadratic formula. By following the tips and tricks outlined in this article, you can become proficient in solving for $x$ in quadratic equations and apply this technique to a wide range of mathematical problems.

Related Problems

If you are looking for more practice problems on solving for $x$ in quadratic equations, here are a few related problems:

• Solve for $x$: $(2x + 3)(x - 4) = 0$
• Solve for $x$: $(x + 2)(3x - 1) = 0$
• Solve for $x$: $(x - 1)(2x + 5) = 0$

These problems require the same techniques as the original problem, and they are a great way to practice solving for $x$ in quadratic equations.

Tips and Tricks

Here are a few tips and tricks to help you solve for $x$ in quadratic equations:

• Always use the zero-product property to find the solutions.
• Set each factor equal to zero and solve for $x$.
• Use algebraic techniques, such as adding, subtracting, multiplying, and dividing, to isolate the variable $x$.
• Verify your solutions by plugging them back into the original equation.
• Use a calculator to check your solutions and verify that they satisfy the equation.

By following these tips and tricks, you can become proficient in solving for $x$ in quadratic equations and apply this technique to a wide range of mathematical problems.