Complete The Square To Solve The Equation Below.$x^2 + X = \frac{11}{4}$A. $x = 3 ; X = -1$ B. $x = 1 + \sqrt{3} ; X = 1 - \sqrt{3}$ C. $x = -2 + \sqrt{3} ; X = -2 - \sqrt{3}$ D. $x = -\frac{1}{2} + \sqrt{3} ;

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Introduction

In algebra, completing the square is a technique used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. This method involves manipulating the equation to express it in the form (x+d)2=e(x + d)^2 = e, where dd and ee are constants. By doing so, we can easily find the solutions to the equation. In this article, we will use the technique of completing the square to solve the equation x2+x=114x^2 + x = \frac{11}{4}.

Understanding the Equation

The given equation is x2+x=114x^2 + x = \frac{11}{4}. To solve this equation, we need to isolate the variable xx. However, the equation is not in the standard form of a quadratic equation, which is ax2+bx+c=0ax^2 + bx + c = 0. To make it easier to solve, we can rewrite the equation as x2+x−114=0x^2 + x - \frac{11}{4} = 0.

Completing the Square

To complete the square, we need to add and subtract a constant term to the left-hand side of the equation. The constant term we need to add is (b2)2\left(\frac{b}{2}\right)^2, where bb is the coefficient of the xx term. In this case, b=1b = 1, so we need to add (12)2=14\left(\frac{1}{2}\right)^2 = \frac{1}{4} to the left-hand side of the equation.

x^2 + x - \frac{11}{4} = 0
x^2 + x + \frac{1}{4} = \frac{11}{4} + \frac{1}{4}
(x + \frac{1}{2})^2 = \frac{12}{4}
(x + \frac{1}{2})^2 = 3

Solving for x

Now that we have completed the square, we can easily solve for xx. We can take the square root of both sides of the equation to get:

x + \frac{1}{2} = \pm \sqrt{3}
x = -\frac{1}{2} \pm \sqrt{3}

Simplifying the Solutions

The solutions to the equation are x=−12+3x = -\frac{1}{2} + \sqrt{3} and x=−12−3x = -\frac{1}{2} - \sqrt{3}. These are the values of xx that satisfy the equation x2+x=114x^2 + x = \frac{11}{4}.

Conclusion

In this article, we used the technique of completing the square to solve the equation x2+x=114x^2 + x = \frac{11}{4}. We added and subtracted a constant term to the left-hand side of the equation to make it easier to solve. By completing the square, we were able to easily find the solutions to the equation. The solutions to the equation are x=−12+3x = -\frac{1}{2} + \sqrt{3} and x=−12−3x = -\frac{1}{2} - \sqrt{3}.

Final Answer

The final answer to the equation x2+x=114x^2 + x = \frac{11}{4} is:

  • x=−12+3x = -\frac{1}{2} + \sqrt{3}
  • x=−12−3x = -\frac{1}{2} - \sqrt{3}

These are the values of xx that satisfy the equation.

Discussion

The technique of completing the square is a powerful tool for solving quadratic equations. It allows us to easily find the solutions to the equation by manipulating the equation to express it in the form (x+d)2=e(x + d)^2 = e. By completing the square, we can solve equations that would be difficult or impossible to solve using other methods.

Example Problems

Here are some example problems that you can try to practice completing the square:

  • x2+4x=12x^2 + 4x = 12
  • x2−6x=9x^2 - 6x = 9
  • x2+2x=6x^2 + 2x = 6

Try to complete the square and solve for xx in each of these equations.

Tips and Tricks

Here are some tips and tricks to help you complete the square:

  • Make sure to add and subtract the same constant term to the left-hand side of the equation.
  • Use the formula (b2)2\left(\frac{b}{2}\right)^2 to find the constant term to add.
  • Simplify the equation as much as possible before completing the square.
  • Check your work by plugging the solutions back into the original equation.

By following these tips and tricks, you can become proficient in completing the square and solving quadratic equations.

Conclusion

In conclusion, completing the square is a powerful technique for solving quadratic equations. By manipulating the equation to express it in the form (x+d)2=e(x + d)^2 = e, we can easily find the solutions to the equation. With practice and patience, you can become proficient in completing the square and solving quadratic equations.

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Introduction

Completing the square is a powerful technique for solving quadratic equations. However, it can be a challenging concept to grasp, especially for those who are new to algebra. In this article, we will provide a Q&A guide to help you understand the concept of completing the square and how to apply it to solve quadratic equations.

Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It involves manipulating the equation to express it in the form (x+d)2=e(x + d)^2 = e, where dd and ee are constants.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

  1. Rewrite the equation in the form ax2+bx+c=0ax^2 + bx + c = 0.
  2. Add and subtract a constant term to the left-hand side of the equation. The constant term is (b2)2\left(\frac{b}{2}\right)^2, where bb is the coefficient of the xx term.
  3. Simplify the equation as much as possible.
  4. Take the square root of both sides of the equation to solve for xx.

Q: What is the formula for completing the square?

A: The formula for completing the square is:

(x+b2)2=(b2)2+c\left(x + \frac{b}{2}\right)^2 = \left(\frac{b}{2}\right)^2 + c

Q: How do I choose the correct value for dd?

A: To choose the correct value for dd, you need to follow these steps:

  1. Identify the coefficient of the xx term, which is bb.
  2. Divide bb by 2 to get b2\frac{b}{2}.
  3. Add b2\frac{b}{2} to the left-hand side of the equation to complete the square.

Q: What are some common mistakes to avoid when completing the square?

A: Here are some common mistakes to avoid when completing the square:

  • Not adding and subtracting the same constant term to the left-hand side of the equation.
  • Not simplifying the equation as much as possible.
  • Not taking the square root of both sides of the equation to solve for xx.
  • Not checking the solutions by plugging them back into the original equation.

Q: How do I check my work when completing the square?

A: To check your work when completing the square, you need to follow these steps:

  1. Plug the solutions back into the original equation to make sure they are true.
  2. Simplify the equation as much as possible to make sure it is in the correct form.
  3. Check that the equation is in the form (x+d)2=e(x + d)^2 = e, where dd and ee are constants.

Q: What are some real-world applications of completing the square?

A: Completing the square has many real-world applications, including:

  • Solving quadratic equations in physics and engineering.
  • Finding the maximum or minimum value of a quadratic function.
  • Modeling population growth and decline.
  • Solving optimization problems.

Conclusion

In conclusion, completing the square is a powerful technique for solving quadratic equations. By following the steps outlined in this article, you can become proficient in completing the square and solving quadratic equations. Remember to avoid common mistakes and check your work to ensure that your solutions are correct.

Final Tips

Here are some final tips to help you master completing the square:

  • Practice, practice, practice! The more you practice, the more comfortable you will become with completing the square.
  • Use online resources and video tutorials to help you understand the concept.
  • Join a study group or find a study buddy to help you stay motivated and learn from others.
  • Don't be afraid to ask for help if you are struggling with a particular concept.

By following these tips and practicing regularly, you can become a master of completing the square and solving quadratic equations.