Consider The Quadratic Equation $x^2 - 20x + 13 = 0$. Completing The Square Leads To The Equivalent Equation:$(x - \square)^2 = \square$ Type The Correct Answer In Each Box. Use Numerals Instead Of Words.

Introduction

The quadratic equation is a fundamental concept in mathematics, and solving it is crucial in various fields such as physics, engineering, and computer science. In this article, we will focus on solving the quadratic equation $x^2 - 20x + 13 = 0$ using the method of completing the square. This method involves rewriting the quadratic equation in a form that allows us to easily identify the solutions.

Completing the Square Method

The completing the square method is a technique used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The basic idea behind this method is to rewrite the quadratic equation in a form that allows us to easily identify the solutions. This is done by manipulating the equation to create a perfect square trinomial on the left-hand side.

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

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

$$x^2 - 20x + 13 = 0$$

Subtract 13 from both sides:

$$x^2 - 20x = -13$$

Step 2: Add and Subtract the Square of Half the Coefficient of x

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

The coefficient of x is -20, so half of this is -10. The square of -10 is 100.

$$x^2 - 20x + 100 - 100 = -13$$

Step 3: Rewrite the Left-Hand Side as a Perfect Square Trinomial

Now that we have added and subtracted the square of half the coefficient of x, we can rewrite the left-hand side of the equation as a perfect square trinomial.

$$(x - 10)^2 - 100 = -13$$

Step 4: Add 100 to Both Sides

The final step is to add 100 to both sides of the equation to isolate the perfect square trinomial on the left-hand side.

$$(x - 10)^2 = 87$$

Solving for x

Now that we have the equation in the form $(x - \square)^2 = \square$, we can easily identify the solutions. The solutions are the values of x that make the equation true.

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

$$x - 10 = \pm \sqrt{87}$$

Adding 10 to both sides gives us:

$$x = 10 \pm \sqrt{87}$$

Conclusion

In this article, we have solved the quadratic equation $x^2 - 20x + 13 = 0$ using the method of completing the square. We have shown that the equation can be rewritten in the form $(x - \square)^2 = \square$, which allows us to easily identify the solutions. The solutions are $x = 10 \pm \sqrt{87}$.

Discussion

The method of completing the square is a powerful tool for solving quadratic equations. It allows us to rewrite the equation in a form that makes it easy to identify the solutions. This method is particularly useful when the quadratic equation has a complex solution.

In conclusion, the method of completing the square is a fundamental concept in mathematics that has numerous applications in various fields. It is a powerful tool for solving quadratic equations and is an essential part of any mathematician's toolkit.

References

• [1] "Completing the Square" by Math Open Reference
• [2] "Quadratic Equations" by Khan Academy
• [3] "Completing the Square" by Purplemath

Further Reading

• "Quadratic Equations and Functions" by Math Is Fun
• "Completing the Square" by IXL
• "Quadratic Equations" by Wolfram MathWorld
  Quadratic Equation Completing the Square Q&A =============================================

Introduction

In our previous article, we discussed how to solve quadratic equations using the method of completing the square. This method involves rewriting the quadratic equation in a form that allows us to easily identify the solutions. 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 method used to solve quadratic equations of the form $ax^2 + bx + c = 0$. It involves rewriting the quadratic equation in a form that allows us to easily identify the solutions.

Q: Why is completing the square useful?

A: Completing the square is useful because it allows us to rewrite the quadratic equation in a form that makes it easy to identify the solutions. This method is particularly useful when the quadratic equation has a complex solution.

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 $ax^2 + bx + c = 0$ and you want to find the solutions.

Q: What are the steps to complete the square?

A: The steps to complete the square 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 x to the left-hand side of the equation.
3. Rewrite the left-hand side as a perfect square trinomial.
4. Add the square of half the coefficient of x to both sides of the equation.

Q: What is the formula for completing the square?

A: The formula for completing the square is:

$$(x - \frac{b}{2})^2 = (\frac{b}{2})^2 - c$$

Q: How do I solve for x?

A: To solve for x, you need to take the square root of both sides of the equation.

Q: What are the solutions to the quadratic equation?

A: The solutions to the quadratic equation are the values of x that make the equation true.

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

A: Yes, you can use completing the square to solve quadratic equations with complex solutions.

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

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

• Not moving the constant term to the right-hand side of the equation.
• Not adding and subtracting the square of half the coefficient of x to the left-hand side of the equation.
• Not rewriting the left-hand side as a perfect square trinomial.
• Not adding the square of half the coefficient of x to both sides of the equation.

Conclusion

In this article, we have answered some frequently asked questions about completing the square. We have discussed the steps to complete the square, the formula for completing the square, and some common mistakes to avoid. Completing the square is a powerful tool for solving quadratic equations, and it is an essential part of any mathematician's toolkit.

References

• [1] "Completing the Square" by Math Open Reference
• [2] "Quadratic Equations" by Khan Academy
• [3] "Completing the Square" by Purplemath

Further Reading

• "Quadratic Equations and Functions" by Math Is Fun
• "Completing the Square" by IXL
• "Quadratic Equations" by Wolfram MathWorld