Select The Correct Answer.The Equation Can Be Solved By Completing The Square. What Number Should Go In The Blanks For The First Step?$\[ X^2 - 18x + \_ = 4 + \_ \\]A. 324 B. 81 C. 9 D. 18
Introduction
Completing the square is a powerful method for solving quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can then be easily solved. In this article, we will explore how to complete the square and apply it to solve a quadratic equation.
What is Completing the Square?
Completing the square is a technique used to solve quadratic equations of the form ax^2 + bx + c = 0. The method involves rewriting the equation in a perfect square form, which can be easily solved. The process involves adding and subtracting a constant term to create a perfect square trinomial.
Step 1: Identify the Coefficients
To complete the square, we need to identify the coefficients of the quadratic equation. In the given equation, x^2 - 18x + _ = 4 + _, we can see that the coefficient of x^2 is 1, the coefficient of x is -18, and the constant term is 4 + _. We need to find the number that should go in the blanks for the first step.
Step 2: Find the Number to Complete the Square
To complete the square, we need to find the number that should be added to the left-hand side of the equation to make it a perfect square trinomial. The number to be added is the square of half the coefficient of x. In this case, the coefficient of x is -18, so half of it is -9. The square of -9 is 81.
Step 3: Add and Subtract the Number
We add 81 to the left-hand side of the equation and subtract 81 from the right-hand side to maintain the equality. This gives us:
x^2 - 18x + 81 = 4 + 81
Step 4: Simplify the Equation
We can simplify the equation by combining like terms:
x^2 - 18x + 81 = 85
Conclusion
In conclusion, to complete the square, we need to find the number that should be added to the left-hand side of the equation to make it a perfect square trinomial. In this case, the number is 81. By adding and subtracting 81, we can rewrite the equation in a perfect square form, which can be easily solved.
Answer
The correct answer is B. 81.
Additional Tips
- To complete the square, we need to identify the coefficients of the quadratic equation.
- We need to find the number that should be added to the left-hand side of the equation to make it a perfect square trinomial.
- The number to be added is the square of half the coefficient of x.
- We add and subtract the number to maintain the equality.
Examples
- Solve the equation x^2 + 6x + _ = 9 by completing the square.
- Solve the equation x^2 - 12x + _ = 16 by completing the square.
Real-World Applications
Completing the square has many real-world applications, including:
- Solving quadratic equations in physics and engineering
- Finding the maximum or minimum value of a quadratic function
- Modeling population growth and decay
Conclusion
Introduction
Completing the square is a powerful method for solving quadratic equations. In our previous article, we explored how to complete the square and apply it to solve a quadratic equation. 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 solve quadratic equations of the form ax^2 + bx + c = 0. The method involves rewriting the equation in a perfect square form, which can be easily solved.
Q: How do I know if I can complete the square?
A: You can complete the square if the coefficient of x^2 is 1. If the coefficient of x^2 is not 1, you may need to divide the entire equation by the coefficient of x^2 to make it 1.
Q: What is the first step in completing the square?
A: The first step in completing the square is to identify the coefficients of the quadratic equation. You need to find the coefficient of x^2, the coefficient of x, and the constant term.
Q: How do I find the number to complete the square?
A: To find the number to complete the square, you need to find the square of half the coefficient of x. The number to be added is the square of half the coefficient of x.
Q: What is the formula for finding the number to complete the square?
A: The formula for finding the number to complete the square is:
number = (coefficient of x / 2)^2
Q: Can I complete the square if the coefficient of x is negative?
A: Yes, you can complete the square if the coefficient of x is negative. The process is the same as if the coefficient of x were positive.
Q: How do I add and subtract the number to complete the square?
A: To add and subtract the number to complete the square, you need to add the number to the left-hand side of the equation and subtract the number from the right-hand side to maintain the equality.
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. The process is the same as if the coefficients were real numbers.
Q: What are some real-world applications of completing the square?
A: Completing the square has many real-world applications, including:
- Solving quadratic equations in physics and engineering
- Finding the maximum or minimum value of a quadratic function
- Modeling population growth and decay
Q: Can I use completing the square to solve quadratic equations with rational coefficients?
A: Yes, you can use completing the square to solve quadratic equations with rational coefficients. The process is the same as if the coefficients were real numbers.
Conclusion
In conclusion, completing the square is a powerful method for solving quadratic equations. By answering these frequently asked questions, we hope to have provided you with a better understanding of how to complete the square and apply it to solve quadratic equations.
Additional Tips
- Make sure to identify the coefficients of the quadratic equation before starting the process.
- Find the number to complete the square by using the formula (coefficient of x / 2)^2.
- Add and subtract the number to complete the square to maintain the equality.
- Use completing the square to solve quadratic equations with complex coefficients.
Examples
- Solve the equation x^2 + 6x + _ = 9 by completing the square.
- Solve the equation x^2 - 12x + _ = 16 by completing the square.
Real-World Applications
Completing the square has many real-world applications, including:
- Solving quadratic equations in physics and engineering
- Finding the maximum or minimum value of a quadratic function
- Modeling population growth and decay