А)4с3-32= Ә)а2+4аb+4b2-3a2b-6ab2=
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Introduction
Algebraic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore two algebraic equations and provide a step-by-step guide on how to solve them.
Equation 1: 4x^3 - 32 = 0
Understanding the Equation
The first equation is a cubic equation in the form of 4x^3 - 32 = 0. To solve this equation, we need to isolate the variable x.
Step 1: Factor Out the Greatest Common Factor (GCF)
The first step in solving this equation is to factor out the greatest common factor (GCF) of the terms. In this case, the GCF is 4.
4x^3 - 32 = 4(x^3 - 8) = 0
Step 2: Factor the Quadratic Expression
Next, we need to factor the quadratic expression x^3 - 8. We can do this by recognizing that x^3 - 8 is a difference of cubes.
x^3 - 8 = (x - 2)(x^2 + 2x + 4)
Step 3: Set Each Factor Equal to Zero
Now that we have factored the quadratic expression, we can set each factor equal to zero and solve for x.
4(x - 2)(x^2 + 2x + 4) = 0
x - 2 = 0 or x^2 + 2x + 4 = 0
Step 4: Solve for x
Solving for x, we get:
x = 2 or x^2 + 2x + 4 = 0
The quadratic expression x^2 + 2x + 4 does not have real solutions, so we can ignore it.
Conclusion
The solutions to the equation 4x^3 - 32 = 0 are x = 2.
Equation 2: a^2 + 4ab + 4b^2 - 3a^2b - 6ab^2 = 0
Understanding the Equation
The second equation is a quadratic equation in the form of a^2 + 4ab + 4b^2 - 3a^2b - 6ab^2 = 0. To solve this equation, we need to isolate the variables a and b.
Step 1: Factor Out the Greatest Common Factor (GCF)
The first step in solving this equation is to factor out the greatest common factor (GCF) of the terms. In this case, the GCF is a^2b.
a^2b(a + 4b - 3ab - 6b^2) = 0
Step 2: Factor the Quadratic Expression
Next, we need to factor the quadratic expression a + 4b - 3ab - 6b^2. We can do this by recognizing that a + 4b - 3ab - 6b^2 is a quadratic expression in the form of ax^2 + bx + c.
a + 4b - 3ab - 6b^2 = (a - 2b)(a - 3b)
Step 3: Set Each Factor Equal to Zero
Now that we have factored the quadratic expression, we can set each factor equal to zero and solve for a and b.
a^2b(a - 2b)(a - 3b) = 0
a - 2b = 0 or a - 3b = 0
Step 4: Solve for a and b
Solving for a and b, we get:
a = 2b or a = 3b
Conclusion
The solutions to the equation a^2 + 4ab + 4b^2 - 3a^2b - 6ab^2 = 0 are a = 2b and a = 3b.
Conclusion
In this article, we have solved two algebraic equations using a step-by-step guide. The first equation was a cubic equation in the form of 4x^3 - 32 = 0, and the second equation was a quadratic equation in the form of a^2 + 4ab + 4b^2 - 3a^2b - 6ab^2 = 0. We have shown that by factoring out the greatest common factor, factoring the quadratic expression, and setting each factor equal to zero, we can solve these equations and find the values of the variables.
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Introduction
Algebraic equations can be a challenging topic for many students and professionals. In this article, we will answer some frequently asked questions (FAQs) about algebraic equations to help you better understand this concept.
Q: What is an algebraic equation?
A: An algebraic equation is a mathematical statement that contains variables and constants, and is equal to zero. It is a way of expressing a relationship between variables and constants.
Q: What are the different types of algebraic equations?
A: There are several types of algebraic equations, including:
- Linear equations: These are equations in which the highest power of the variable is 1. For example, 2x + 3 = 0.
- Quadratic equations: These are equations in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0.
- Cubic equations: These are equations in which the highest power of the variable is 3. For example, x^3 - 8 = 0.
- Polynomial equations: These are equations in which the highest power of the variable is greater than 3. For example, x^4 + 2x^3 + 3x^2 + 4x + 5 = 0.
Q: How do I solve an algebraic equation?
A: To solve an algebraic equation, you need to isolate the variable on one side of the equation. This can be done by using various techniques such as factoring, combining like terms, and using inverse operations.
Q: What is factoring in algebra?
A: Factoring is a technique used to simplify an algebraic expression by expressing it as a product of simpler expressions. For example, the expression x^2 + 4x + 4 can be factored as (x + 2)(x + 2).
Q: What is the difference between a variable and a constant?
A: A variable is a symbol that represents a value that can change. For example, x is a variable. A constant is a value that does not change. For example, 2 is a constant.
Q: How do I determine the degree of an algebraic equation?
A: The degree of an algebraic equation is the highest power of the variable in the equation. For example, the equation x^2 + 4x + 4 has a degree of 2.
Q: What is the importance of algebraic equations in real-life situations?
A: Algebraic equations are used in a wide range of real-life situations, including physics, engineering, economics, and computer science. They are used to model and solve problems in these fields.
Q: Can algebraic equations be used to solve problems in other areas of mathematics?
A: Yes, algebraic equations can be used to solve problems in other areas of mathematics, including geometry and trigonometry.
Q: How do I know if an algebraic equation has a solution?
A: To determine if an algebraic equation has a solution, you need to check if the equation is consistent. If the equation is consistent, then it has a solution.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation in which the highest power of the variable is 1. A quadratic equation is an equation in which the highest power of the variable is 2.
Q: Can algebraic equations be used to solve problems in other areas of science?
A: Yes, algebraic equations can be used to solve problems in other areas of science, including physics and engineering.
Conclusion
In this article, we have answered some frequently asked questions (FAQs) about algebraic equations. We have covered topics such as the definition of an algebraic equation, the different types of algebraic equations, and how to solve them. We have also discussed the importance of algebraic equations in real-life situations and how they can be used to solve problems in other areas of mathematics and science.