Select The Correct Answer.What Are The Solutions To This Equation?$7x^2 - 28 = 0$A. $x = -2$ And $x = 2$B. $x = -4$ And $x = 4$C. $x = -\sqrt{7}$ And $x = \sqrt{7}$D. $x =

by ADMIN 172 views

=====================================================

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. We will use the given equation 7x2−28=07x^2 - 28 = 0 as an example to demonstrate the steps involved in solving quadratic equations.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, xx) is two. The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

where aa, bb, and cc are constants. The value of aa cannot be zero, as this would result in a linear equation rather than a quadratic one.

The Given Equation

The given equation is:

7x2−28=07x^2 - 28 = 0

To solve this equation, we need to isolate the variable xx. We can start by adding 28 to both sides of the equation to get:

7x2=287x^2 = 28

Solving for xx

Now that we have the equation 7x2=287x^2 = 28, we can solve for xx by dividing both sides by 7:

x2=287x^2 = \frac{28}{7}

Simplifying the right-hand side, we get:

x2=4x^2 = 4

Taking the Square Root

To solve for xx, we need to take the square root of both sides of the equation:

x=±4x = \pm \sqrt{4}

Simplifying the right-hand side, we get:

x=±2x = \pm 2

The Solutions

Therefore, the solutions to the equation 7x2−28=07x^2 - 28 = 0 are:

x=−2 and x=2x = -2 \text{ and } x = 2

Conclusion

In this article, we have demonstrated the steps involved in solving quadratic equations. We have used the given equation 7x2−28=07x^2 - 28 = 0 as an example to show how to isolate the variable xx and solve for its value. The solutions to the equation are x=−2x = -2 and x=2x = 2.

Answer Key

The correct answer is:

  • A. x=−2x = -2 and x=2x = 2

This is the only option that matches the solutions we obtained in the previous section.

Final Thoughts

Solving quadratic equations is an essential skill for students and professionals alike. By following the steps outlined in this article, you can solve quadratic equations with ease. Remember to always check your solutions by plugging them back into the original equation to ensure that they are correct.

Additional Resources

If you are struggling with quadratic equations or need additional practice, there are many online resources available to help you. Some popular resources include:

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

These resources can provide you with additional practice and help you to better understand the concepts involved in solving quadratic equations.

Common Mistakes

When solving quadratic equations, there are several common mistakes to avoid. Some of these mistakes include:

  • Not checking solutions: Always check your solutions by plugging them back into the original equation to ensure that they are correct.
  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when solving quadratic equations.
  • Not simplifying expressions: Simplify expressions as much as possible to make it easier to solve the equation.

By avoiding these common mistakes, you can ensure that you are solving quadratic equations correctly and efficiently.

Real-World Applications

Quadratic equations have many real-world applications. Some examples include:

  • Physics: Quadratic equations are used to model 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.

By understanding how to solve quadratic equations, you can apply this knowledge to a wide range of real-world problems.

Conclusion

In conclusion, solving quadratic equations is an essential skill for students and professionals alike. By following the steps outlined in this article, you can solve quadratic equations with ease. Remember to always check your solutions by plugging them back into the original equation to ensure that they are correct. With practice and patience, you can become proficient in solving quadratic equations and apply this knowledge to a wide range of real-world problems.

=====================================================

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. 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, xx) is two. The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

where aa, bb, and cc are constants. The value of aa cannot be zero, as this would result in a linear equation rather than a quadratic one.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to isolate the variable xx. You can start by adding or subtracting a constant to both sides of the equation to get:

ax2+bx+c=kax^2 + bx + c = k

where kk is a constant. Then, you can use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

to find the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the solutions to a quadratic equation. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic equation.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula:

b2−4acb^2 - 4ac

If the discriminant is positive, the equation has two distinct 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: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you need to examine the discriminant. If the discriminant is:

  • Positive: The equation has two distinct real solutions.
  • Zero: The equation has one real solution.
  • Negative: The equation has no real solutions.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is:

ax+b=0ax + b = 0

where aa and bb are constants.

Q: Can I use the quadratic formula to solve a linear equation?

A: No, you cannot use the quadratic formula to solve a linear equation. The quadratic formula is used to solve quadratic equations, while linear equations can be solved using simple algebraic manipulations.

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

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

  • Not checking solutions: Always check your solutions by plugging them back into the original equation to ensure that they are correct.
  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when solving quadratic equations.
  • Not simplifying expressions: Simplify expressions as much as possible to make it easier to solve the equation.

Q: How do I apply quadratic equations to real-world problems?

A: Quadratic equations have many real-world applications. Some examples include:

  • Physics: Quadratic equations are used to model 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.

By understanding how to solve quadratic equations, you can apply this knowledge to a wide range of real-world problems.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve quadratic equations with ease. Remember to always check your solutions by plugging them back into the original equation to ensure that they are correct. With practice and patience, you can become proficient in solving quadratic equations and apply this knowledge to a wide range of real-world problems.

Additional Resources

If you are struggling with quadratic equations or need additional practice, there are many online resources available to help you. Some popular resources include:

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

These resources can provide you with additional practice and help you to better understand the concepts involved in solving quadratic equations.

Real-World Applications

Quadratic equations have many real-world applications. Some examples include:

  • Physics: Quadratic equations are used to model 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.

By understanding how to solve quadratic equations, you can apply this knowledge to a wide range of real-world problems.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve quadratic equations with ease. Remember to always check your solutions by plugging them back into the original equation to ensure that they are correct. With practice and patience, you can become proficient in solving quadratic equations and apply this knowledge to a wide range of real-world problems.