Select The Correct Answer From Each Drop-down Menu.Using The Distributive Property To Factorize The Equation 3 X 2 + 24 X = 0 3x^2 + 24x = 0 3 X 2 + 24 X = 0 , You Get □ \square □ .The Solution Of The Equation Is □ \square □ .

Introduction

In this article, we will explore the concept of using the distributive property to factorize a quadratic equation. We will apply this concept to the given equation $3x^2 + 24x = 0$ and find its solution.

The Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand a product of two or more expressions. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

Applying the Distributive Property to the Given Equation

To factorize the given equation $3x^2 + 24x = 0$, we can use the distributive property. We can rewrite the equation as:

$$3x^2 + 24x = 3x(x + 8) = 0$$

Solving the Equation

Now that we have factorized the equation, we can solve for $x$. We can set each factor equal to zero and solve for $x$:

$$3x = 0 \Rightarrow x = 0$$

$$x + 8 = 0 \Rightarrow x = -8$$

The Solution of the Equation

Therefore, the solution of the equation $3x^2 + 24x = 0$ is $x = 0$ or $x = -8$.

Conclusion

In this article, we used the distributive property to factorize the quadratic equation $3x^2 + 24x = 0$. We then solved for $x$ by setting each factor equal to zero. The solution of the equation is $x = 0$ or $x = -8$.

Key Takeaways

• The distributive property is a fundamental concept in algebra that allows us to expand a product of two or more expressions.
• We can use the distributive property to factorize a quadratic equation.
• To solve a quadratic equation, we can set each factor equal to zero and solve for the variable.

Practice Problems

1. Factorize the quadratic equation $2x^2 + 14x = 0$ using the distributive property.
2. Solve the equation $x^2 + 6x = 0$ by setting each factor equal to zero.
3. Factorize the quadratic equation $4x^2 + 20x = 0$ using the distributive property.

Answer Key

1. $2x(x + 7) = 0$
2. $x = 0$ or $x = -6$
3. $4x(x + 5) = 0$

Final Thoughts

Q: What is the distributive property in algebra?

A: The distributive property is a fundamental concept in algebra that allows us to expand a product of two or more expressions. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

Q: How do I apply the distributive property to factorize a quadratic equation?

A: To factorize a quadratic equation using the distributive property, you can follow these steps:

1. Identify the quadratic equation and rewrite it in the form $ax^2 + bx + c = 0$.
2. Look for two numbers whose product is $ac$ and whose sum is $b$. These numbers are the factors of the quadratic expression.
3. Rewrite the quadratic expression as a product of two binomials using the factors you found in step 2.
4. Set each binomial equal to zero and solve for $x$.

Q: What are some common mistakes to avoid when using the distributive property to factorize quadratic equations?

A: Some common mistakes to avoid when using the distributive property to factorize quadratic equations include:

• Not identifying the correct factors of the quadratic expression.
• Not rewriting the quadratic expression as a product of two binomials correctly.
• Not setting each binomial equal to zero and solving for $x$ correctly.

Q: Can I use the distributive property to factorize any quadratic equation?

A: Yes, you can use the distributive property to factorize any quadratic equation. However, not all quadratic equations can be factored using the distributive property. Some quadratic equations may require other methods, such as the quadratic formula, to solve.

Q: How do I know if a quadratic equation can be factored using the distributive property?

A: To determine if a quadratic equation can be factored using the distributive property, you can try the following:

• Look for two numbers whose product is $ac$ and whose sum is $b$. If you can find these numbers, then the quadratic equation can be factored using the distributive property.
• Try rewriting the quadratic expression as a product of two binomials using the distributive property. If you can rewrite it correctly, then the quadratic equation can be factored using the distributive property.

Q: What are some real-world applications of using the distributive property to factorize quadratic equations?

A: Some real-world applications of using the distributive property to factorize quadratic equations include:

• Solving systems of linear equations.
• Finding the maximum or minimum value of a quadratic function.
• Modeling real-world phenomena, such as the motion of an object under the influence of gravity.

Q: Can I use the distributive property to factorize quadratic equations with complex coefficients?

A: Yes, you can use the distributive property to factorize quadratic equations with complex coefficients. However, you will need to use complex numbers and complex arithmetic to solve the equation.

Q: How do I factorize a quadratic equation with a negative coefficient?

A: To factorize a quadratic equation with a negative coefficient, you can follow these steps:

1. Rewrite the quadratic equation with a positive coefficient by multiplying both sides of the equation by $-1$.
2. Factorize the quadratic equation using the distributive property.
3. Rewrite the factored form of the quadratic equation with a negative coefficient.

Q: Can I use the distributive property to factorize quadratic equations with fractional coefficients?

A: Yes, you can use the distributive property to factorize quadratic equations with fractional coefficients. However, you will need to use fractional arithmetic to solve the equation.

Q: How do I factorize a quadratic equation with a coefficient of 1?

A: To factorize a quadratic equation with a coefficient of 1, you can follow these steps:

1. Rewrite the quadratic equation in the form $x^2 + bx + c = 0$.
2. Look for two numbers whose product is $c$ and whose sum is $b$. These numbers are the factors of the quadratic expression.
3. Rewrite the quadratic expression as a product of two binomials using the factors you found in step 2.
4. Set each binomial equal to zero and solve for $x$.

Q: Can I use the distributive property to factorize quadratic equations with a coefficient of 0?

A: Yes, you can use the distributive property to factorize quadratic equations with a coefficient of 0. However, you will need to use the fact that $0x = 0$ to solve the equation.

Q: How do I factorize a quadratic equation with a coefficient of 0 and a constant term of 0?

A: To factorize a quadratic equation with a coefficient of 0 and a constant term of 0, you can follow these steps:

1. Rewrite the quadratic equation in the form $ax^2 = 0$.
2. Set each factor equal to zero and solve for $x$.

Q: Can I use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of 1?

A: Yes, you can use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of 1. However, you will need to use the fact that $0x = 0$ to solve the equation.

Q: How do I factorize a quadratic equation with a coefficient of 0 and a constant term of 1?

A: To factorize a quadratic equation with a coefficient of 0 and a constant term of 1, you can follow these steps:

1. Rewrite the quadratic equation in the form $x^2 = 1$.
2. Set each factor equal to zero and solve for $x$.

Q: Can I use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of -1?

A: Yes, you can use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of -1. However, you will need to use the fact that $0x = 0$ to solve the equation.

Q: How do I factorize a quadratic equation with a coefficient of 0 and a constant term of -1?

A: To factorize a quadratic equation with a coefficient of 0 and a constant term of -1, you can follow these steps:

1. Rewrite the quadratic equation in the form $x^2 = -1$.
2. Set each factor equal to zero and solve for $x$.

Q: Can I use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of a fraction?

A: Yes, you can use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of a fraction. However, you will need to use fractional arithmetic to solve the equation.

Q: How do I factorize a quadratic equation with a coefficient of 0 and a constant term of a fraction?

A: To factorize a quadratic equation with a coefficient of 0 and a constant term of a fraction, you can follow these steps:

1. Rewrite the quadratic equation in the form $ax^2 = b$.
2. Set each factor equal to zero and solve for $x$.

Q: Can I use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of a complex number?

A: Yes, you can use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of a complex number. However, you will need to use complex numbers and complex arithmetic to solve the equation.

Q: How do I factorize a quadratic equation with a coefficient of 0 and a constant term of a complex number?

A: To factorize a quadratic equation with a coefficient of 0 and a constant term of a complex number, you can follow these steps:

1. Rewrite the quadratic equation in the form $ax^2 = b$.
2. Set each factor equal to zero and solve for $x$.

Q: Can I use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of a variable?

A: Yes, you can use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of a variable. However, you will need to use variable arithmetic to solve the equation.

Q: How do I factorize a quadratic equation with a coefficient of 0 and a constant term of a variable?

A: To factorize a quadratic equation with a coefficient of 0 and a constant term of a variable, you can follow these steps:

1. Rewrite the quadratic equation in the form $ax^2 = b$.
2. Set each factor equal to zero and solve for $x$.

Q: Can I use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of a function?

A: Yes, you can use the distributive property to factorize quadratic equations with a coefficient of 0 and a constant term of a function. However, you will need to use function