Solve For X. 3x^2+7=82 Need Help? Watch This Video! Responses ±7 ±7 ±10 ±10 ±5 ±5 ±6 ASAP PLEASE

Mar 13, 2025 by ADMIN 97 views

Solve for x: 3x^2 + 7 = 82 - A Step-by-Step Guide

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Are you struggling to solve quadratic equations? Do you need help with algebraic manipulations? Look no further! In this article, we will guide you through the process of solving the quadratic equation 3x^2 + 7 = 82. We will break down the solution into manageable steps, and provide you with a clear understanding of the concepts involved.

Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. In our equation, 3x^2 + 7 = 82, we can rewrite it in the standard form as 3x^2 - 75 = 0.

The Importance of Quadratic Equations

Quadratic equations have numerous applications in various fields

Quadratic equations have a wide range of applications in physics, engineering, economics, and computer science. They are used to model real-world phenomena, such as the trajectory of a projectile, the vibration of a spring, or the growth of a population. In mathematics, quadratic equations are used to solve problems involving optimization, geometry, and algebra.

Step 1: Subtract 7 from Both Sides

To solve the equation 3x^2 - 75 = 0, we need to isolate the term with the variable (x). The first step is to subtract 7 from both sides of the equation. This will give us 3x^2 = 82 - 7.

# Subtract 7 from both sides
equation = "3x^2 - 75 = 0"
new_equation = "3x^2 = 82 - 7"
print(new_equation)

Step 2: Simplify the Right-Hand Side

Now that we have 3x^2 = 75, we can simplify the right-hand side by performing the subtraction.

# Simplify the right-hand side
new_equation = "3x^2 = 75"
print(new_equation)

Step 3: Divide Both Sides by 3

To isolate the term with the variable (x), we need to divide both sides of the equation by 3. This will give us x^2 = 25.

# Divide both sides by 3
new_equation = "x^2 = 25"
print(new_equation)

Step 4: Take the Square Root of Both Sides

Now that we have x^2 = 25, we can take the square root of both sides to solve for x. This will give us x = ±√25.

# Take the square root of both sides
import math
x = math.sqrt(25)
print(x)

Step 5: Simplify the Square Root

The square root of 25 is 5. Therefore, x = ±5.

# Simplify the square root
x = 5
print(x)

Step 6: Write the Final Solution

The final solution to the equation 3x^2 + 7 = 82 is x = ±5.

# Write the final solution
final_solution = "x = ±5"
print(final_solution)

Conclusion

In this article, we have solved the quadratic equation 3x^2 + 7 = 82 using a step-by-step approach. We have broken down the solution into manageable steps, and provided you with a clear understanding of the concepts involved. We have also used Python code to illustrate each step of the solution. With this guide, you should be able to solve quadratic equations with confidence.

Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two.

How do I solve a quadratic equation?

To solve a quadratic equation, you need to isolate the term with the variable (x) by performing algebraic manipulations such as addition, subtraction, multiplication, and division.

What is the importance of quadratic equations?

Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and computer science.

Additional Resources

Video Tutorials

If you need help with solving quadratic equations, you can watch video tutorials on YouTube or other online platforms.

Online Resources

You can also use online resources such as Khan Academy, Mathway, or Wolfram Alpha to solve quadratic equations.

Practice Problems

To practice solving quadratic equations, you can try solving problems on websites such as IXL, Math Open Reference, or Brilliant.

Final Thoughts

Solving quadratic equations is a skill that requires practice

Solving quadratic equations is a skill that requires practice and patience. With this guide, you should be able to solve quadratic equations with confidence. Remember to practice regularly to improve your skills.

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Are you struggling to solve quadratic equations? Do you have questions about quadratic equations? Look no further! In this article, we will answer some of the most frequently asked questions about quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to isolate the term with the variable (x) by performing algebraic manipulations such as addition, subtraction, multiplication, and division. You can use the quadratic formula, factoring, or completing the square to solve a quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. The formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

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

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (in this case, x) is one. A quadratic equation, on the other hand, is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two.

Q: Can I use a calculator to solve a quadratic equation?

A: 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 quadratic equations.

Q: What is the importance of quadratic equations?

A: Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and computer science. They are used to model real-world phenomena, such as the trajectory of a projectile, the vibration of a spring, or the growth of a population.

Q: Can I use quadratic equations to solve problems in real life?

A: Yes, you can use quadratic equations to solve problems in real life. Quadratic equations are used to model real-world phenomena, such as the trajectory of a projectile, the vibration of a spring, or the growth of a population.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

Not following the order of operations
Not simplifying the equation
Not checking the solutions
Not using the correct formula

Q: Can I use quadratic equations to solve systems of equations?

A: Yes, you can use quadratic equations to solve systems of equations. Quadratic equations can be used to solve systems of linear equations, as well as systems of nonlinear equations.

Q: What are some real-world applications of quadratic equations?

A: Some real-world applications of quadratic equations include:

Modeling the trajectory of a projectile
Modeling the vibration of a spring
Modeling the growth of a population
Modeling the motion of an object under the influence of gravity

Q: Can I use quadratic equations to solve optimization problems?

A: Yes, you can use quadratic equations to solve optimization problems. Quadratic equations can be used to find the maximum or minimum value of a function.

Q: What are some common types of quadratic equations?

A: Some common types of quadratic equations include:

Quadratic equations with real roots
Quadratic equations with complex roots
Quadratic equations with no real roots
Quadratic equations with repeated roots

Q: Can I use quadratic equations to solve problems in physics?

A: Yes, you can use quadratic equations to solve problems in physics. Quadratic equations are used to model the motion of objects under the influence of gravity, as well as the vibration of springs.

Q: What are some common mistakes to avoid when graphing quadratic equations?

A: Some common mistakes to avoid when graphing quadratic equations include:

Not using the correct scale
Not labeling the axes
Not including the vertex
Not including the roots

Q: Can I use quadratic equations to solve problems in engineering?

A: Yes, you can use quadratic equations to solve problems in engineering. Quadratic equations are used to model the motion of objects under the influence of gravity, as well as the vibration of springs.

Q: What are some common types of quadratic equations in engineering?

A: Some common types of quadratic equations in engineering include:

Quadratic equations used to model the motion of objects under the influence of gravity
Quadratic equations used to model the vibration of springs
Quadratic equations used to model the growth of populations
Quadratic equations used to model the motion of objects under the influence of friction

Q: Can I use quadratic equations to solve problems in economics?

A: Yes, you can use quadratic equations to solve problems in economics. Quadratic equations are used to model the growth of populations, as well as the motion of objects under the influence of gravity.

Q: What are some common types of quadratic equations in economics?

A: Some common types of quadratic equations in economics include:

Quadratic equations used to model the growth of populations
Quadratic equations used to model the motion of objects under the influence of gravity
Quadratic equations used to model the vibration of springs
Quadratic equations used to model the motion of objects under the influence of friction

Q: Can I use quadratic equations to solve problems in computer science?

A: Yes, you can use quadratic equations to solve problems in computer science. Quadratic equations are used to model the growth of populations, as well as the motion of objects under the influence of gravity.

Q: What are some common types of quadratic equations in computer science?

A: Some common types of quadratic equations in computer science include:

Quadratic equations used to model the growth of populations
Quadratic equations used to model the motion of objects under the influence of gravity
Quadratic equations used to model the vibration of springs
Quadratic equations used to model the motion of objects under the influence of friction

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic equations. We have covered topics such as the definition of a quadratic equation, the quadratic formula, and the importance of quadratic equations. We have also discussed some common mistakes to avoid when solving quadratic equations, as well as some real-world applications of quadratic equations. With this guide, you should be able to solve quadratic equations with confidence.