(7m − 5)(6m + 5) = 0 Write Your Answers As Integers Or As Proper Or Improper Fractions In Simplest Form. M = Or M = Scratchpad
Introduction
In algebra, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. In this article, we will solve the quadratic equation (7m − 5)(6m + 5) = 0.
Understanding the Equation
The given equation is a product of two binomials, (7m − 5) and (6m + 5). To solve this equation, we need to find the values of m that make the product equal to zero. This means that either (7m − 5) = 0 or (6m + 5) = 0.
Solving the First Factor
Let's start by solving the first factor, (7m − 5) = 0.
Step 1: Add 5 to both sides of the equation
7m − 5 + 5 = 0 + 5
Step 2: Simplify the equation
7m = 5
Step 3: Divide both sides of the equation by 7
m = 5/7
So, the value of m that makes the first factor equal to zero is m = 5/7.
Solving the Second Factor
Now, let's solve the second factor, (6m + 5) = 0.
Step 1: Subtract 5 from both sides of the equation
6m + 5 - 5 = 0 - 5
Step 2: Simplify the equation
6m = -5
Step 3: Divide both sides of the equation by 6
m = -5/6
So, the value of m that makes the second factor equal to zero is m = -5/6.
Conclusion
In conclusion, the values of m that make the equation (7m − 5)(6m + 5) = 0 are m = 5/7 and m = -5/6. These values satisfy the equation and make the product equal to zero.
Discussion
The equation (7m − 5)(6m + 5) = 0 is a quadratic equation that can be solved by factoring. The solution involves finding the values of m that make the product of the two binomials equal to zero. In this case, we found two values of m, m = 5/7 and m = -5/6, that satisfy the equation.
Example Use Cases
The equation (7m − 5)(6m + 5) = 0 can be used in various real-world applications, such as:
- Physics: In physics, the equation can be used to model the motion of an object under the influence of a force. For example, the equation can be used to find the velocity of an object at a given time.
- Engineering: In engineering, the equation can be used to design and optimize systems, such as electrical circuits or mechanical systems.
- Computer Science: In computer science, the equation can be used to solve problems in algorithms and data structures.
Tips and Tricks
When solving quadratic equations, it's essential to remember the following tips and tricks:
- Factorization: Factorization is a powerful technique for solving quadratic equations. It involves expressing the quadratic expression as a product of two binomials.
- Simplification: Simplification is an essential step in solving quadratic equations. It involves combining like terms and eliminating any unnecessary variables.
- Checking: Checking is a crucial step in solving quadratic equations. It involves verifying that the solution satisfies the original equation.
Common Mistakes
When solving quadratic equations, it's essential to avoid the following common mistakes:
- Incorrect factorization: Incorrect factorization can lead to incorrect solutions. Make sure to factorize the quadratic expression correctly.
- Simplification errors: Simplification errors can lead to incorrect solutions. Make sure to simplify the equation correctly.
- Checking errors: Checking errors can lead to incorrect solutions. Make sure to check the solution correctly.
Conclusion
Q: What is the quadratic equation (7m − 5)(6m + 5) = 0?
A: The quadratic equation (7m − 5)(6m + 5) = 0 is a product of two binomials, (7m − 5) and (6m + 5). To solve this equation, we need to find the values of m that make the product equal to zero.
Q: How do I solve the quadratic equation (7m − 5)(6m + 5) = 0?
A: To solve the quadratic equation (7m − 5)(6m + 5) = 0, we need to find the values of m that make the product of the two binomials equal to zero. This means that either (7m − 5) = 0 or (6m + 5) = 0.
Q: What are the values of m that satisfy the equation (7m − 5)(6m + 5) = 0?
A: The values of m that satisfy the equation (7m − 5)(6m + 5) = 0 are m = 5/7 and m = -5/6.
Q: How do I check if the values of m are correct?
A: To check if the values of m are correct, we need to substitute them back into the original equation and verify that the product is equal to zero.
Q: What are some common mistakes to avoid when solving the quadratic equation (7m − 5)(6m + 5) = 0?
A: Some common mistakes to avoid when solving the quadratic equation (7m − 5)(6m + 5) = 0 include:
- Incorrect factorization: Incorrect factorization can lead to incorrect solutions. Make sure to factorize the quadratic expression correctly.
- Simplification errors: Simplification errors can lead to incorrect solutions. Make sure to simplify the equation correctly.
- Checking errors: Checking errors can lead to incorrect solutions. Make sure to check the solution correctly.
Q: What are some real-world applications of the quadratic equation (7m − 5)(6m + 5) = 0?
A: The quadratic equation (7m − 5)(6m + 5) = 0 has various real-world applications, including:
- Physics: In physics, the equation can be used to model the motion of an object under the influence of a force. For example, the equation can be used to find the velocity of an object at a given time.
- Engineering: In engineering, the equation can be used to design and optimize systems, such as electrical circuits or mechanical systems.
- Computer Science: In computer science, the equation can be used to solve problems in algorithms and data structures.
Q: What are some tips and tricks for solving the quadratic equation (7m − 5)(6m + 5) = 0?
A: Some tips and tricks for solving the quadratic equation (7m − 5)(6m + 5) = 0 include:
- Factorization: Factorization is a powerful technique for solving quadratic equations. It involves expressing the quadratic expression as a product of two binomials.
- Simplification: Simplification is an essential step in solving quadratic equations. It involves combining like terms and eliminating any unnecessary variables.
- Checking: Checking is a crucial step in solving quadratic equations. It involves verifying that the solution satisfies the original equation.
Q: Can I use technology to solve the quadratic equation (7m − 5)(6m + 5) = 0?
A: Yes, you can use technology to solve the quadratic equation (7m − 5)(6m + 5) = 0. There are various online tools and software programs that can help you solve quadratic equations, including graphing calculators and computer algebra systems.
Q: How do I know if the quadratic equation (7m − 5)(6m + 5) = 0 has a real solution?
A: To determine if the quadratic equation (7m − 5)(6m + 5) = 0 has a real solution, you need to check if the discriminant is positive, zero, or negative. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.
Q: Can I use the quadratic formula to solve the equation (7m − 5)(6m + 5) = 0?
A: Yes, you can use the quadratic formula to solve the equation (7m − 5)(6m + 5) = 0. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation. In this case, a = 42, b = -35, and c = 25. Plugging these values into the quadratic formula, you get m = (35 ± √((-35)^2 - 4(42)(25))) / (2(42)). Simplifying this expression, you get m = (35 ± √(1225 - 4200)) / 84, which simplifies to m = (35 ± √(-2975)) / 84. Since the discriminant is negative, the equation has no real solutions.