There Are Four Steps For Converting The Equation X 2 + Y 2 + 12 X + 2 Y − 1 = 0 X^2+y^2+12x+2y-1=0 X 2 + Y 2 + 12 X + 2 Y − 1 = 0 Into Standard Form By Completing The Square. Complete The Last Step.1. Group The X X X Terms Together And The Y Y Y Terms Together, And Move The Constant Term

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Introduction

Completing the square is a powerful technique used to convert quadratic equations into standard form. This method involves manipulating the equation to express it in a perfect square trinomial form, which can be easily factored or solved. In this article, we will focus on converting the equation x2+y2+12x+2y−1=0x^2+y^2+12x+2y-1=0 into standard form by completing the square.

Step 1: Group the xx Terms Together and the yy Terms Together, and Move the Constant Term

The first step in completing the square is to group the xx terms together and the yy terms together, and move the constant term to the right-hand side of the equation. This can be done as follows:

x2+12x+y2+2y−1=0x^2 + 12x + y^2 + 2y - 1 = 0

Grouping the xx terms together and the yy terms together, we get:

(x2+12x)+(y2+2y)−1=0(x^2 + 12x) + (y^2 + 2y) - 1 = 0

Moving the constant term to the right-hand side, we get:

(x2+12x)+(y2+2y)=1(x^2 + 12x) + (y^2 + 2y) = 1

Step 2: Add and Subtract the Square of Half the Coefficient of xx and the Square of Half the Coefficient of yy

The next step is to add and subtract the square of half the coefficient of xx and the square of half the coefficient of yy to the left-hand side of the equation. This will create a perfect square trinomial.

For the xx terms, the coefficient is 12, so half of this is 6. The square of 6 is 36.

For the yy terms, the coefficient is 2, so half of this is 1. The square of 1 is 1.

Adding and subtracting these values, we get:

(x2+12x+36)+(y2+2y+1)−36−1=1(x^2 + 12x + 36) + (y^2 + 2y + 1) - 36 - 1 = 1

Simplifying, we get:

(x2+12x+36)+(y2+2y+1)=38(x^2 + 12x + 36) + (y^2 + 2y + 1) = 38

Step 3: Factor the Perfect Square Trinomials

The next step is to factor the perfect square trinomials on the left-hand side of the equation.

For the xx terms, we can factor the perfect square trinomial as follows:

x2+12x+36=(x+6)2x^2 + 12x + 36 = (x + 6)^2

For the yy terms, we can factor the perfect square trinomial as follows:

y2+2y+1=(y+1)2y^2 + 2y + 1 = (y + 1)^2

Substituting these values, we get:

(x+6)2+(y+1)2=38(x + 6)^2 + (y + 1)^2 = 38

Step 4: Complete the Square

The final step is to complete the square by expressing the equation in standard form.

We can do this by rearranging the equation as follows:

(x+6)2+(y+1)2=38(x + 6)^2 + (y + 1)^2 = 38

This is the standard form of the equation, where the left-hand side is a perfect square trinomial and the right-hand side is a constant.

Conclusion

In this article, we have completed the square to convert the equation x2+y2+12x+2y−1=0x^2+y^2+12x+2y-1=0 into standard form. We have followed the four steps of completing the square, which involve grouping the xx terms together and the yy terms together, adding and subtracting the square of half the coefficient of xx and the square of half the coefficient of yy, factoring the perfect square trinomials, and completing the square. The final result is the standard form of the equation, which can be easily factored or solved.

Final Answer

The final answer is:

Introduction

Completing the square is a powerful technique used to convert quadratic equations into standard form. In our previous article, we walked through the four steps of completing the square to convert the equation x2+y2+12x+2y−1=0x^2+y^2+12x+2y-1=0 into standard form. In this article, we will answer some frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a technique used to convert quadratic equations into standard form. It involves manipulating the equation to express it in a perfect square trinomial form, which can be easily factored or solved.

Q: Why do we need to complete the square?

A: We need to complete the square to convert quadratic equations into standard form, which can be easily factored or solved. This is useful in many areas of mathematics, such as algebra, geometry, and calculus.

Q: What are the four steps of completing the square?

A: The four steps of completing the square are:

  1. Group the xx terms together and the yy terms together, and move the constant term to the right-hand side of the equation.
  2. Add and subtract the square of half the coefficient of xx and the square of half the coefficient of yy to the left-hand side of the equation.
  3. Factor the perfect square trinomials on the left-hand side of the equation.
  4. Complete the square by expressing the equation in standard form.

Q: How do I know when to complete the square?

A: You should complete the square when you have a quadratic equation that cannot be easily factored or solved. This is often the case when the equation has a non-zero constant term.

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

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

  • Forgetting to add and subtract the square of half the coefficient of xx and the square of half the coefficient of yy.
  • Factoring the perfect square trinomials incorrectly.
  • Not completing the square by expressing the equation in standard form.

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

A: Yes, you can use completing the square to solve quadratic equations with complex coefficients. However, you will need to use complex numbers and complex arithmetic.

Q: Are there any other techniques for solving quadratic equations?

A: Yes, there are other techniques for solving quadratic equations, including:

  • Factoring: This involves expressing the quadratic equation as a product of two binomials.
  • Quadratic formula: This involves using a formula to find the solutions to the quadratic equation.
  • Graphing: This involves graphing the quadratic equation on a coordinate plane and finding the solutions.

Conclusion

In this article, we have answered some frequently asked questions about completing the square. We have discussed the four steps of completing the square, common mistakes to avoid, and other techniques for solving quadratic equations. We hope this article has been helpful in understanding completing the square.

Final Answer

The final answer is:

Completing the square is a powerful technique used to convert quadratic equations into standard form. It involves manipulating the equation to express it in a perfect square trinomial form, which can be easily factored or solved.

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