Solve By Completing The Square:${ Y^2 + 28y = -41 } W R I T E Y O U R A N S W E R S A S I N T E G E R S , P R O P E R O R I M P R O P E R F R A C T I O N S I N S I M P L E S T F O R M , O R D E C I M A L S R O U N D E D T O T H E N E A R E S T H U N D R E D T H . Write Your Answers As Integers, Proper Or Improper Fractions In Simplest Form, Or Decimals Rounded To The Nearest Hundredth. W R I T Eyo U R An S W Ers A S In T E G Ers , P Ro P Eror Im P Ro P Er F R A C T I O N S In S Im Pl Es T F Or M , Or D Ec Ima L Sro U N D E D T O T H E N E A Res T H U N D Re D T H . { Y = \square \quad \text{or} \quad Y = \square \}

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Introduction

Completing the square is a powerful algebraic technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square trinomial form, which can be easily solved. In this article, we will focus on solving the quadratic equation y2+28y=−41y^2 + 28y = -41 by completing the square.

What is Completing the Square?

Completing the square is a method of solving quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. The process involves rewriting the equation in a perfect square trinomial form, which can be easily solved. The general form of a perfect square trinomial is (x+d)2=x2+2dx+d2(x + d)^2 = x^2 + 2dx + d^2.

Step 1: Move the Constant Term to the Right Side

To complete the square, we need to move the constant term to the right side of the equation. We can do this by subtracting −41-41 from both sides of the equation.

y2+28y=−41y^2 + 28y = -41

Subtracting −41-41 from both sides gives us:

y2+28y+41=0y^2 + 28y + 41 = 0

Step 2: Find the Value to Add to Both Sides

To complete the square, we need to find the value to add to both sides of the equation. This value is equal to the square of half the coefficient of the yy term. In this case, the coefficient of the yy term is 2828, so we need to find the square of half of 2828.

(282)2=142=196\left(\frac{28}{2}\right)^2 = 14^2 = 196

Step 3: Add the Value to Both Sides

We add the value we found in the previous step to both sides of the equation.

y2+28y+196=41+196y^2 + 28y + 196 = 41 + 196

Step 4: Factor the Left Side

The left side of the equation is now a perfect square trinomial. We can factor it as follows:

(y+14)2=237(y + 14)^2 = 237

Step 5: Solve for y

To solve for yy, we take the square root of both sides of the equation.

y+14=±237y + 14 = \pm \sqrt{237}

Subtracting 1414 from both sides gives us:

y=−14±237y = -14 \pm \sqrt{237}

Conclusion

In this article, we solved the quadratic equation y2+28y=−41y^2 + 28y = -41 by completing the square. We moved the constant term to the right side, found the value to add to both sides, added the value to both sides, factored the left side, and finally solved for yy. The solutions to the equation are y=−14±237y = -14 \pm \sqrt{237}.

Final Answer

The final answer is −14±237\boxed{-14 \pm \sqrt{237}}.

Discussion

Completing the square is a powerful algebraic technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square trinomial form, which can be easily solved. In this article, we focused on solving the quadratic equation y2+28y=−41y^2 + 28y = -41 by completing the square. We moved the constant term to the right side, found the value to add to both sides, added the value to both sides, factored the left side, and finally solved for yy. The solutions to the equation are y=−14±237y = -14 \pm \sqrt{237}.

Related Topics

  • Solving quadratic equations by factoring
  • Solving quadratic equations by using the quadratic formula
  • Solving quadratic equations by graphing

References

Additional Resources

Note: The final answer is in the format of a boxed expression, which is a common way to present the final answer in mathematics. The discussion section provides additional context and related topics, while the references section lists relevant resources for further learning. The additional resources section provides links to online tools and platforms that can be used to supplement learning.

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Introduction

Completing the square is a powerful algebraic technique used to solve quadratic equations. In our previous article, we provided a step-by-step guide on how to complete the square to solve the quadratic equation y2+28y=−41y^2 + 28y = -41. In this article, we will answer some frequently asked questions about completing the square.

Q&A

Q: What is completing the square?

A: Completing the square is a method of solving quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It involves manipulating the equation to express it in a perfect square trinomial form, which can be easily solved.

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

A: You should use completing the square when the quadratic equation cannot be easily factored or when the quadratic formula is not applicable.

Q: What is the main advantage of completing the square?

A: The main advantage of completing the square is that it allows us to solve quadratic equations without having to use the quadratic formula.

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

A: No, completing the square is only applicable to quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It is not applicable to quadratic equations of the form ax2+c=0ax^2 + c = 0.

Q: How do I find the value to add to both sides of the equation?

A: To find the value to add to both sides of the equation, you need to find the square of half the coefficient of the yy term.

Q: What is the final step in completing the square?

A: The final step in completing the square is to solve for yy by taking the square root of both sides of the equation.

Common Mistakes

Mistake 1: Not moving the constant term to the right side

A: Make sure to move the constant term to the right side of the equation before completing the square.

Mistake 2: Not finding the value to add to both sides

A: Make sure to find the value to add to both sides of the equation before adding it.

Mistake 3: Not adding the value to both sides

A: Make sure to add the value to both sides of the equation before factoring the left side.

Tips and Tricks

Tip 1: Use a calculator to check your answers

A: Use a calculator to check your answers and make sure they are correct.

Tip 2: Practice, practice, practice

A: Practice completing the square with different types of quadratic equations to become proficient in the technique.

Tip 3: Use visual aids to help you understand the process

A: Use visual aids such as graphs and charts to help you understand the process of completing the square.

Conclusion

Completing the square is a powerful algebraic technique used to solve quadratic equations. In this article, we answered some frequently asked questions about completing the square and provided tips and tricks to help you become proficient in the technique. Remember to practice, practice, practice to become proficient in completing the square.

Final Answer

The final answer is y=−14±237\boxed{y = -14 \pm \sqrt{237}}.

Discussion

Completing the square is a powerful algebraic technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square trinomial form, which can be easily solved. In this article, we focused on answering some frequently asked questions about completing the square and provided tips and tricks to help you become proficient in the technique.

Related Topics

  • Solving quadratic equations by factoring
  • Solving quadratic equations by using the quadratic formula
  • Solving quadratic equations by graphing

References

Additional Resources

Note: The Q&A section provides answers to frequently asked questions about completing the square, while the common mistakes section highlights common mistakes to avoid. The tips and tricks section provides additional advice and resources to help you become proficient in completing the square.