7(t+12) To The Second Power -20=323
Introduction
In this article, we will delve into the world of quadratic equations and solve the given equation 7(t+12)^2 - 20 = 323. Quadratic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and mathematical techniques. In this discussion, we will break down the equation into manageable steps and provide a clear solution to the problem.
Understanding the Equation
The given equation is 7(t+12)^2 - 20 = 323. To solve this equation, we need to isolate the variable t. The first step is to add 20 to both sides of the equation to get rid of the negative term.
7(t+12)^2 = 323 + 20
7(t+12)^2 = 343
Simplifying the Equation
Next, we need to simplify the equation by dividing both sides by 7.
(t+12)^2 = 343 / 7
(t+12)^2 = 49
Taking the Square Root
Now, we need to take the square root of both sides of the equation to get rid of the squared term.
t+12 = ±√49
t+12 = ±7
Solving for t
Finally, we need to solve for t by subtracting 12 from both sides of the equation.
t = -12 ± 7
t = -12 + 7 or t = -12 - 7
t = -5 or t = -19
Conclusion
In this article, we solved the quadratic equation 7(t+12)^2 - 20 = 323 by breaking it down into manageable steps. We added 20 to both sides of the equation, simplified it by dividing both sides by 7, took the square root of both sides, and finally solved for t. The solutions to the equation are t = -5 and t = -19.
Step-by-Step Solution
Here is the step-by-step solution to the equation:
- Add 20 to both sides of the equation: 7(t+12)^2 = 323 + 20
- Simplify the equation by dividing both sides by 7: (t+12)^2 = 343 / 7
- Take the square root of both sides of the equation: t+12 = ±√49
- Solve for t by subtracting 12 from both sides of the equation: t = -12 ± 7
Common Mistakes
When solving quadratic equations, it's easy to make mistakes. Here are some common mistakes to avoid:
- Not adding 20 to both sides of the equation
- Not simplifying the equation by dividing both sides by 7
- Not taking the square root of both sides of the equation
- Not solving for t by subtracting 12 from both sides of the equation
Real-World Applications
Quadratic equations have many real-world applications. Here are a few examples:
- Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity.
- Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
- Computer Science: Quadratic equations are used in algorithms and data structures to solve problems efficiently.
Final Thoughts
Introduction
In our previous article, we solved the quadratic equation 7(t+12)^2 - 20 = 323. In this article, we will answer some frequently asked questions about quadratic equations and provide additional insights into solving them.
Q: What is a quadratic equation?
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 ax^2 + bx + c = 0, where a, b, and c are constants.
A: How do I solve a quadratic equation?
To solve a quadratic equation, you can use the following steps:
- Add or subtract the same value to both sides of the equation to isolate the squared term.
- Simplify the equation by dividing both sides by the coefficient of the squared term.
- Take the square root of both sides of the equation to get rid of the squared term.
- Solve for the variable by adding or subtracting the same value to both sides of the equation.
Q: What is the difference between a quadratic equation and a linear equation?
A linear equation is a polynomial equation of degree one, which means the highest power of the variable is one. It is typically written in the form ax + b = 0, where a and b are constants.
A: How do I know if a quadratic equation has real or complex solutions?
To determine if a quadratic equation has real or complex solutions, you can use the discriminant, which is the expression b^2 - 4ac. 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 two complex solutions.
Q: Can I use a calculator to solve a quadratic equation?
Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that can be used to solve the equation.
Q: What are some common mistakes to avoid when solving quadratic equations?
Some common mistakes to avoid when solving quadratic equations include:
- Not adding or subtracting the same value to both sides of the equation
- Not simplifying the equation by dividing both sides by the coefficient of the squared term
- Not taking the square root of both sides of the equation
- Not solving for the variable by adding or subtracting the same value to both sides of the equation
Q: How do I apply quadratic equations to real-world problems?
Quadratic equations have many real-world applications, including:
- Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity.
- Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
- Computer Science: Quadratic equations are used in algorithms and data structures to solve problems efficiently.
Q: Can I use quadratic equations to solve systems of equations?
Yes, you can use quadratic equations to solve systems of equations. By solving the quadratic equation, you can find the values of the variables that satisfy the system of equations.
Conclusion
In this article, we answered some frequently asked questions about quadratic equations and provided additional insights into solving them. Quadratic equations are a fundamental concept in mathematics, and understanding them is essential for success in mathematics and other fields. By following the steps outlined in this article, you can solve quadratic equations and apply them to real-world problems.
Additional Resources
For more information on quadratic equations, you can consult the following resources:
- Khan Academy: Quadratic Equations
- Mathway: Quadratic Equations
- Wolfram Alpha: Quadratic Equations
Final Thoughts
In conclusion, quadratic equations are a powerful tool for solving problems in mathematics and other fields. By understanding the concepts and techniques outlined in this article, you can solve quadratic equations and apply them to real-world problems.