Solve The Quadratic Equation By Completing The Square.${ \begin{array}{l} 0 = X^2 - 6x + 4 \ -4 = X^2 - 6x \ -4 + 9 = (x^2 - 6x + 9) \ 5 = (x - 3)^2 \end{array} }$What Are The Two Solutions Of The Equation?A. { X = \sqrt{5}$}$

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. One of the methods used to solve quadratic equations is by completing the square. This method involves manipulating the equation to express it in a perfect square form, which can then be easily solved. In this article, we will explore how to solve a quadratic equation by completing the square, using the equation 0=x2−6x+40 = x^2 - 6x + 4 as an example.

Understanding the Method of Completing the Square

The method of completing the square is a technique used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. The basic idea behind this method is to manipulate the equation to express it in the form (x+d)2=e(x + d)^2 = e, where dd and ee are constants. This can be done by adding and subtracting a constant term to the equation, which allows us to create a perfect square trinomial.

Step-by-Step Solution

Let's apply the method of completing the square to the equation 0=x2−6x+40 = x^2 - 6x + 4. The first step is to move the constant term to the right-hand side of the equation, so that we have:

x2−6x=−4x^2 - 6x = -4

Next, 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 the square of half the coefficient of the xx term, which is (−6/2)2=9(-6/2)^2 = 9. Adding and subtracting 9 to the left-hand side of the equation gives us:

x2−6x+9=−4+9x^2 - 6x + 9 = -4 + 9

Simplifying the right-hand side of the equation, we get:

(x−3)2=5(x - 3)^2 = 5

Solving for x

Now that we have expressed the equation in the form (x−d)2=e(x - d)^2 = e, we can easily solve for xx. Taking the square root of both sides of the equation, we get:

x−3=±5x - 3 = \pm \sqrt{5}

Adding 3 to both sides of the equation, we get:

x=3±5x = 3 \pm \sqrt{5}

Therefore, the two solutions of the equation are x=3+5x = 3 + \sqrt{5} and x=3−5x = 3 - \sqrt{5}.

Conclusion

In this article, we have seen how to solve a quadratic equation by completing the square. This method involves manipulating the equation to express it in a perfect square form, which can then be easily solved. By following the step-by-step solution outlined in this article, we can solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. The two solutions of the equation 0=x2−6x+40 = x^2 - 6x + 4 are x=3+5x = 3 + \sqrt{5} and x=3−5x = 3 - \sqrt{5}.

Example Problems

Here are a few example problems that you can try to practice the method of completing the square:

  • 0=x2+4x+40 = x^2 + 4x + 4
  • 0=x2−2x−30 = x^2 - 2x - 3
  • 0=x2+2x−60 = x^2 + 2x - 6

Tips and Tricks

Here are a few tips and tricks that you can use to help you solve quadratic equations by completing the square:

  • Make sure to move the constant term to the right-hand side of the equation before adding and subtracting the constant term.
  • Use the formula (x−d)2=e(x - d)^2 = e to help you remember the steps involved in completing the square.
  • Check your work by plugging the solutions back into the original equation to make sure they are true.

Common Mistakes

Here are a few common mistakes that you can avoid when solving quadratic equations by completing the square:

  • Don't forget to move the constant term to the right-hand side of the equation before adding and subtracting the constant term.
  • Make sure to add and subtract the correct constant term to the left-hand side of the equation.
  • Check your work by plugging the solutions back into the original equation to make sure they are true.

Real-World Applications

Quadratic equations have many real-world applications, including:

  • 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.
  • Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.

Conclusion

Frequently Asked Questions

Q: What is completing the square?

A: Completing the square is a method used to solve quadratic equations by manipulating the equation to express it in a perfect square form. This involves adding and subtracting a constant term to the left-hand side of the equation, which allows us to create a perfect square trinomial.

Q: How do I know when to use completing the square?

A: You should use completing the square when the quadratic equation is in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. This method is particularly useful when the equation cannot be easily factored.

Q: What are the steps involved in completing the square?

A: The steps involved in completing the square are:

  1. Move the constant term to the right-hand side of the equation.
  2. Add and subtract a constant term to the left-hand side of the equation.
  3. Simplify the right-hand side of the equation.
  4. Take the square root of both sides of the equation.
  5. Add or subtract the constant term from both sides of the equation.

Q: How do I know which constant term to add and subtract?

A: The constant term you need to add and subtract is the square of half the coefficient of the xx term. This is calculated by taking half of the coefficient of the xx term and squaring it.

Q: What if the equation has a negative sign in front of it?

A: If the equation has a negative sign in front of it, you can simply ignore it and proceed with the steps involved in completing the square. The negative sign will be taken care of when you take the square root of both sides of the equation.

Q: Can I use completing the square to solve quadratic equations with complex roots?

A: Yes, you can use completing the square to solve quadratic equations with complex roots. The steps involved in completing the square remain the same, but you will need to take the square root of both sides of the equation and simplify the expression.

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

A: To check your work, you can plug the solutions back into the original equation to make sure they are true. You can also use a calculator to check your work.

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

A: Some common mistakes to avoid when using completing the square include:

  • Forgetting to move the constant term to the right-hand side of the equation.
  • Adding and subtracting the wrong constant term to the left-hand side of the equation.
  • Not taking the square root of both sides of the equation.
  • Not checking your work by plugging the solutions back into the original equation.

Q: Can I use completing the square to solve quadratic equations with rational roots?

A: Yes, you can use completing the square to solve quadratic equations with rational roots. The steps involved in completing the square remain the same, but you will need to simplify the expression and take the square root of both sides of the equation.

Q: How do I know if a quadratic equation has rational roots?

A: A quadratic equation has rational roots if the discriminant is a perfect square. The discriminant is calculated by using the formula b2−4acb^2 - 4ac, where aa, bb, and cc are the coefficients of the quadratic equation.

Conclusion

In conclusion, completing the square is a powerful technique used to solve quadratic equations. By following the steps involved in completing the square, you can solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. Remember to check your work by plugging the solutions back into the original equation to make sure they are true.