Solve The Equation M 2 − 5 M = 0 M^2 - 5m = 0 M 2 − 5 M = 0 .A) { 5 , − 5 } \{5, -5\} { 5 , − 5 } B) { − 5 , 0 } \{-5, 0\} { − 5 , 0 } C) { − 6 , 3 } \{-6, 3\} { − 6 , 3 } D) { 5 , 0 } \{5, 0\} { 5 , 0 }

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Introduction

Solving equations is a fundamental concept in mathematics, and it is essential to understand how to approach different types of equations. In this article, we will focus on solving a quadratic equation of the form $m^2 - 5m = 0$. We will use algebraic methods to find the solutions to this equation and compare them with the given options.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = 1$, $b = -5$, and $c = 0$. To solve this equation, we need to find the values of $m$ that satisfy the equation.

Factoring the Equation

One way to solve the equation is to factor it. We can rewrite the equation as $m(m - 5) = 0$. This tells us that either $m = 0$ or $m - 5 = 0$. We can solve for $m$ in both cases.

Solving for $m$

If $m = 0$, then we have found one solution to the equation. If $m - 5 = 0$, then we can add 5 to both sides of the equation to get $m = 5$. Therefore, the solutions to the equation are $m = 0$ and $m = 5$.

Comparing with the Options

Now that we have found the solutions to the equation, we can compare them with the given options. The options are:

A) $\{5, -5\}$ B) $\{-5, 0\}$ C) $\{-6, 3\}$ D) $\{5, 0\}$

We can see that the solutions we found, $m = 0$ and $m = 5$, match option D) $\{5, 0\}$. Therefore, the correct answer is option D.

Conclusion

In this article, we solved the equation $m^2 - 5m = 0$ using algebraic methods. We factored the equation and solved for $m$ to find the solutions. We then compared the solutions with the given options and found that the correct answer is option D) $\{5, 0\}$.

Final Answer

The final answer is option D) $\{5, 0\}$.

Additional Tips and Tricks

• When solving quadratic equations, it is essential to factor the equation if possible.
• If the equation cannot be factored, you can use the quadratic formula to find the solutions.
• Always check your solutions by plugging them back into the original equation.

Common Mistakes to Avoid

• When factoring the equation, make sure to distribute the terms correctly.
• When solving for $m$, make sure to check for extraneous solutions.
• When comparing the solutions with the options, make sure to read the options carefully and choose the correct answer.

Real-World Applications

Solving quadratic equations has many real-world applications, such as:

• Finding the maximum or minimum value of a quadratic function
• Determining the vertex of a parabola
• Modeling population growth or decline
• Solving problems in physics, engineering, and economics

Practice Problems

Here are some practice problems to help you reinforce your understanding of solving quadratic equations:

1. Solve the equation $x^2 + 4x + 4 = 0$.
2. Solve the equation $y^2 - 6y + 9 = 0$.
3. Solve the equation $z^2 + 2z + 1 = 0$.

Solutions to Practice Problems

1. The solutions to the equation $x^2 + 4x + 4 = 0$ are $x = -2$ and $x = -2$.
2. The solutions to the equation $y^2 - 6y + 9 = 0$ are $y = 3$ and $y = 3$.
3. The solutions to the equation $z^2 + 2z + 1 = 0$ are $z = -1$ and $z = -1$.

Conclusion

Solving quadratic equations is an essential skill in mathematics, and it has many real-world applications. In this article, we solved the equation $m^2 - 5m = 0$ using algebraic methods and compared the solutions with the given options. We also provided some practice problems and solutions to help you reinforce your understanding of solving quadratic equations.

Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand how to approach different types of equations. In this article, we will provide a Q&A section to help you reinforce your understanding of solving quadratic equations.

Q1: What is a quadratic equation?

A1: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q2: How do I solve a quadratic equation?

A2: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The method you choose will depend on the specific equation and your personal preference.

Q3: What is the quadratic formula?

A3: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is written as $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q4: How do I use the quadratic formula?

A4: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and solve for $x$.

Q5: What is the difference between a quadratic equation and a linear equation?

A5: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared variable, while a linear equation does not.

Q6: Can I solve a quadratic equation by graphing?

A6: Yes, you can solve a quadratic equation by graphing. By graphing the quadratic function, you can find the x-intercepts, which represent the solutions to the equation.

Q7: What is the vertex of a quadratic function?

A7: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is also the minimum or maximum value of the function.

Q8: How do I find the vertex of a quadratic function?

A8: To find the vertex of a quadratic function, you can use the formula $x = \frac{-b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic equation.

Q9: Can I solve a quadratic equation with complex numbers?

A9: Yes, you can solve a quadratic equation with complex numbers. In fact, complex numbers are often used to solve quadratic equations that have no real solutions.

Q10: What is the significance of solving quadratic equations?

A10: Solving quadratic equations has many real-world applications, such as modeling population growth or decline, determining the vertex of a parabola, and solving problems in physics, engineering, and economics.

Conclusion

Solving quadratic equations is an essential skill in mathematics, and it has many real-world applications. In this article, we provided a Q&A section to help you reinforce your understanding of solving quadratic equations. We hope this article has been helpful in answering your questions and providing you with a better understanding of quadratic equations.

Additional Resources

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

Practice Problems

Here are some practice problems to help you reinforce your understanding of solving quadratic equations:

1. Solve the equation $x^2 + 4x + 4 = 0$.
2. Solve the equation $y^2 - 6y + 9 = 0$.
3. Solve the equation $z^2 + 2z + 1 = 0$.

Solutions to Practice Problems

1. The solutions to the equation $x^2 + 4x + 4 = 0$ are $x = -2$ and $x = -2$.
2. The solutions to the equation $y^2 - 6y + 9 = 0$ are $y = 3$ and $y = 3$.
3. The solutions to the equation $z^2 + 2z + 1 = 0$ are $z = -1$ and $z = -1$.

Conclusion

Solving quadratic equations is an essential skill in mathematics, and it has many real-world applications. In this article, we provided a Q&A section to help you reinforce your understanding of solving quadratic equations. We hope this article has been helpful in answering your questions and providing you with a better understanding of quadratic equations.