Solve By Completing The Square.$\[ H^2 - 16h = 9 \\]Write Your Answers As Integers, Proper Or Improper Fractions In Simplest Form, Or Decimals Rounded To The Nearest Hundredth.$\[ H = \square \\]or $\[ H = \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 factored and solved. In this article, we will focus on solving the quadratic equation h2−16h=9h^2 - 16h = 9 using the completing the square method.

What is Completing the Square?

Completing the square is a method of solving quadratic equations by rewriting them in a perfect square trinomial form. This involves adding and subtracting a constant term to create a perfect square trinomial, which can be factored and solved. The method is based on the concept of a perfect square trinomial, which is a trinomial that can be factored into the square of a binomial.

Step 1: Move the Constant Term

The first step in completing the square is to move the constant term to the right-hand side of the equation. In this case, we have:

h2−16h=9h^2 - 16h = 9

We can move the constant term to the right-hand side by subtracting 9 from both sides:

h2−16h−9=0h^2 - 16h - 9 = 0

Step 2: Add and Subtract a Constant Term

The next step is to add and subtract a constant term to create a perfect square trinomial. The constant term to add is the square of half the coefficient of the linear term. In this case, the coefficient of the linear term is -16, so half of this is -8. The square of -8 is 64, so we add and subtract 64 to the equation:

h2−16h+64−64−9=0h^2 - 16h + 64 - 64 - 9 = 0

Step 3: Factor the Perfect Square Trinomial

The next step is to factor the perfect square trinomial. We can do this by recognizing that the first three terms form a perfect square trinomial:

(h−8)2−73=0(h - 8)^2 - 73 = 0

Step 4: Add 73 to Both Sides

The next step is to add 73 to both sides of the equation to isolate the perfect square trinomial:

(h−8)2=73(h - 8)^2 = 73

Step 5: Take the Square Root

The next step is to take the square root of both sides of the equation. We can do this by recognizing that the square root of a perfect square trinomial is the binomial:

h−8=±73h - 8 = \pm \sqrt{73}

Step 6: Add 8 to Both Sides

The final step is to add 8 to both sides of the equation to solve for h:

h=8±73h = 8 \pm \sqrt{73}

Conclusion

In this article, we have used the completing the square method to solve the quadratic equation h2−16h=9h^2 - 16h = 9. We have moved the constant term to the right-hand side, added and subtracted a constant term to create a perfect square trinomial, factored the perfect square trinomial, added 73 to both sides, taken the square root, and added 8 to both sides to solve for h. The solution is h=8±73h = 8 \pm \sqrt{73}.

Example Problems

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

  • x2+6x=16x^2 + 6x = 16
  • y2−10y=25y^2 - 10y = 25
  • z2+2z=36z^2 + 2z = 36

Tips and Tricks

Here are some tips and tricks to help you use the completing the square method:

  • Make sure to move the constant term to the right-hand side of the equation.
  • Add and subtract a constant term to create a perfect square trinomial.
  • Factor the perfect square trinomial.
  • Add 73 to both sides of the equation.
  • Take the square root of both sides of the equation.
  • Add 8 to both sides of the equation.

Common Mistakes

Here are some common mistakes to avoid when 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 a constant term to create a perfect square trinomial.
  • Failing to factor the perfect square trinomial.
  • Failing to add 73 to both sides of the equation.
  • Failing to take the square root of both sides of the equation.
  • Failing to add 8 to both sides of the equation.

Conclusion

Introduction

In our previous article, we discussed the completing the square method for solving quadratic equations. In this article, we will provide a Q&A section to help you better understand the concept and apply it to different types of problems.

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 trinomial form. This involves adding and subtracting a constant term to create a perfect square trinomial, which can be factored and solved.

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

A: You should use the completing the square method when you have a quadratic equation in the form of ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. The completing the square method is particularly useful when the quadratic equation has a negative discriminant, which means that it has no real solutions.

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

A: The first step in completing the square is to move the constant term to the right-hand side of the equation. This involves subtracting the constant term from both sides of the equation.

Q: How do I add and subtract a constant term to create a perfect square trinomial?

A: To add and subtract a constant term to create a perfect square trinomial, you need to add and subtract the square of half the coefficient of the linear term. The coefficient of the linear term is the number that multiplies the variable, and half of this number is the number that you need to square.

Q: How do I factor the perfect square trinomial?

A: To factor the perfect square trinomial, you need to recognize that it is a perfect square trinomial and factor it accordingly. A perfect square trinomial is a trinomial that can be factored into the square of a binomial.

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

A: The final step in completing the square is to take the square root of both sides of the equation and solve for the variable.

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

A: No, you cannot use the completing the square method to solve all types of quadratic equations. The completing the square method is particularly useful for quadratic equations with a negative discriminant, which means that they have no real solutions.

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 a constant term to create a perfect square trinomial.
  • Failing to factor the perfect square trinomial.
  • Failing to take the square root of both sides of the equation.
  • Failing to add 8 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 try using the completing the square method to solve different types of quadratic equations.

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.

Conclusion

In conclusion, the completing the square method is a powerful algebraic technique used to solve quadratic equations. By following the steps outlined in this article, you can use the completing the square method to solve quadratic equations and improve your algebraic skills. Remember to practice using the completing the square method to become proficient in its use.