Solve For Y Y Y Using The Completing The Square Method: Y 2 − 3 Y + 1 = 0 Y^2 - 3y + 1 = 0 Y 2 − 3 Y + 1 = 0

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. One of the most effective methods for solving quadratic equations is the completing the square method. In this article, we will explore how to use the completing the square method to solve the quadratic equation y2−3y+1=0y^2 - 3y + 1 = 0.

What is the Completing the Square Method?

The completing the square method is a technique used to solve quadratic equations by rewriting them in a perfect square form. This method involves manipulating the quadratic equation to create a perfect square trinomial, which can be factored into the square of a binomial. The completing the square method is a powerful tool for solving quadratic equations, and it has many applications in mathematics, science, and engineering.

Step 1: Move the Constant Term to the Right-Hand Side

To begin solving the quadratic equation y2−3y+1=0y^2 - 3y + 1 = 0 using the completing the square method, we need to move the constant term to the right-hand side of the equation. This will give us:

y2−3y=−1y^2 - 3y = -1

Step 2: Add and Subtract the Square of Half the Coefficient of the Linear Term

Next, we need to add and subtract the square of half the coefficient of the linear term to the left-hand side of the equation. In this case, the coefficient of the linear term is -3, so we need to add and subtract (−32)2=94\left(\frac{-3}{2}\right)^2 = \frac{9}{4}:

y2−3y+94=−1+94y^2 - 3y + \frac{9}{4} = -1 + \frac{9}{4}

Step 3: Factor the Perfect Square Trinomial

Now that we have added and subtracted the square of half the coefficient of the linear term, we can factor the perfect square trinomial on the left-hand side of the equation:

(y−32)2=−1+94(y - \frac{3}{2})^2 = -1 + \frac{9}{4}

Step 4: Simplify the Right-Hand Side

To simplify the right-hand side of the equation, we need to find a common denominator and combine the fractions:

−1+94=−44+94=54-1 + \frac{9}{4} = \frac{-4}{4} + \frac{9}{4} = \frac{5}{4}

So, the equation becomes:

(y−32)2=54(y - \frac{3}{2})^2 = \frac{5}{4}

Step 5: Take the Square Root of Both Sides

To solve for yy, we need to take the square root of both sides of the equation. This will give us:

y−32=±54y - \frac{3}{2} = \pm \sqrt{\frac{5}{4}}

Step 6: Simplify the Square Root

To simplify the square root, we can multiply both the numerator and the denominator by 4:

54=54=52\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{\sqrt{4}} = \frac{\sqrt{5}}{2}

So, the equation becomes:

y−32=±52y - \frac{3}{2} = \pm \frac{\sqrt{5}}{2}

Step 7: Add 32\frac{3}{2} to Both Sides

To solve for yy, we need to add 32\frac{3}{2} to both sides of the equation:

y=32±52y = \frac{3}{2} \pm \frac{\sqrt{5}}{2}

Conclusion

In this article, we have used the completing the square method to solve the quadratic equation y2−3y+1=0y^2 - 3y + 1 = 0. We have moved the constant term to the right-hand side, added and subtracted the square of half the coefficient of the linear term, factored the perfect square trinomial, simplified the right-hand side, taken the square root of both sides, simplified the square root, and finally added 32\frac{3}{2} to both sides to solve for yy. The completing the square method is a powerful tool for solving quadratic equations, and it has many applications in mathematics, science, and engineering.

Example Problems

Here are a few example problems that you can try using the completing the square method:

  • x2+4x+4=0x^2 + 4x + 4 = 0
  • y2−2y+1=0y^2 - 2y + 1 = 0
  • z2+6z+9=0z^2 + 6z + 9 = 0

Tips and Tricks

Here are a few tips and tricks that you can use when solving quadratic equations using the completing the square method:

  • Make sure to move the constant term to the right-hand side of the equation.
  • Add and subtract the square of half the coefficient of the linear term to the left-hand side of the equation.
  • Factor the perfect square trinomial on the left-hand side of the equation.
  • Simplify the right-hand side of the equation.
  • Take the square root of both sides of the equation.
  • Simplify the square root.
  • Add the value that you added to both sides of the equation to both sides of the equation.

Common Mistakes

Here are a few common mistakes that you can make when solving quadratic equations using the completing the square method:

  • Failing to move the constant term to the right-hand side of the equation.
  • Failing to add and subtract the square of half the coefficient of the linear term to the left-hand side of the equation.
  • Failing to factor the perfect square trinomial on the left-hand side of the equation.
  • Failing to simplify the right-hand side of the equation.
  • Failing to take the square root of both sides of the equation.
  • Failing to simplify the square root.
  • Failing to add the value that you added to both sides of the equation to both sides of the equation.

Conclusion

In conclusion, the completing the square method is a powerful tool for solving quadratic equations. By following the steps outlined in this article, you can use the completing the square method to solve quadratic equations and gain a deeper understanding of the underlying mathematics. Remember to move the constant term to the right-hand side of the equation, add and subtract the square of half the coefficient of the linear term to the left-hand side of the equation, factor the perfect square trinomial on the left-hand side of the equation, simplify the right-hand side of the equation, take the square root of both sides of the equation, simplify the square root, and finally add the value that you added to both sides of the equation to both sides of the equation. With practice and patience, you can become proficient in using the completing the square method to solve quadratic equations.

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Q: What is the completing the square method?

A: The completing the square method is a technique used to solve quadratic equations by rewriting them in a perfect square form. This method involves manipulating the quadratic equation to create a perfect square trinomial, which can be factored into the square of a binomial.

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

A: You should use the completing the square method when you are given a quadratic equation in the form of ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and you want to solve for xx.

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

A: The steps involved in the completing the square method are:

  1. Move the constant term to the right-hand side of the equation.
  2. Add and subtract the square of half the coefficient of the linear term to the left-hand side of the equation.
  3. Factor the perfect square trinomial on the left-hand side of the equation.
  4. Simplify the right-hand side of the equation.
  5. Take the square root of both sides of the equation.
  6. Simplify the square root.
  7. Add the value that you added to both sides of the equation to both sides of the equation.

Q: What is the difference between the completing the square method and factoring?

A: The completing the square method and factoring are two different techniques used to solve quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while the completing the square method involves rewriting the quadratic equation in a perfect square form.

Q: When should I use factoring instead of the completing the square method?

A: You should use factoring instead of the completing the square method when the quadratic equation can be easily factored into the product of two binomials.

Q: Can I use the completing the square method to solve all types of quadratic equations?

A: Yes, you can use the completing the square method to solve all types of quadratic equations, including those with complex roots.

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

A: You can determine if a quadratic equation has complex roots by using the discriminant, which is given by the formula b2−4acb^2 - 4ac. If the discriminant is negative, then the quadratic equation has complex roots.

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

A: Yes, you can use the completing the square method to solve quadratic equations with complex roots. However, you will need to use the imaginary unit ii to represent the complex roots.

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

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

  • Failing to move the constant term to the right-hand side of the equation.
  • Failing to add and subtract the square of half the coefficient of the linear term to the left-hand side of the equation.
  • Failing to factor the perfect square trinomial on the left-hand side of the equation.
  • Failing to simplify the right-hand side of the equation.
  • Failing to take the square root of both sides of the equation.
  • Failing to simplify the square root.
  • Failing to add the value that you added to both sides of the equation to both sides of the equation.

Q: How can I practice using the completing the square method?

A: You can practice using the completing the square method by working through example problems and exercises. You can also use online resources and practice tests to help you improve your skills.

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

A: The completing the square method has many real-world applications, including:

  • Solving quadratic equations in physics and engineering.
  • Modeling population growth and decline.
  • Analyzing data and making predictions.
  • Solving optimization problems.

Q: Can I use the completing the square method to solve systems of equations?

A: Yes, you can use the completing the square method to solve systems of equations. However, you will need to use a combination of the completing the square method and other techniques, such as substitution and elimination.

Q: How can I extend my knowledge of the completing the square method to more advanced topics?

A: You can extend your knowledge of the completing the square method to more advanced topics by studying advanced algebra and calculus. You can also explore applications of the completing the square method in fields such as physics, engineering, and computer science.