In The Expression, The Value That Was Added To Create The Perfect Square Trinomial Is Being Multiplied By -16, So The Value That Should Be Subtracted From 15 To Maintain Equivalency Is ____.Complete The Table To Complete The Square Without Using
Introduction
Completing the square is a powerful algebraic technique used to solve quadratic equations and express them in a perfect square trinomial form. This method involves manipulating the equation to create a perfect square trinomial, which can be factored into the square of a binomial. In this article, we will explore the concept of completing the square and provide a step-by-step guide on how to complete the square without using the formula.
What is Completing the Square?
Completing the square is a process of transforming a quadratic equation into a perfect square trinomial form. This is achieved by adding and subtracting a constant term to the equation, which allows us to factor the equation into the square of a binomial. The constant term added is called the "value to be added" or "value to be subtracted," depending on whether it is added or subtracted from the original equation.
The Formula for Completing the Square
The formula for completing the square is:
x^2 + bx + c = (x + b/2)^2 - (b/2)^2 + c
However, in this article, we will not use this formula. Instead, we will complete the square by manipulating the equation using algebraic techniques.
Step 1: Identify the Quadratic Equation
To complete the square, we need to identify the quadratic equation. The equation is in the form of ax^2 + bx + c = 0, where a, b, and c are constants.
Step 2: Move the Constant Term to the Right-Hand Side
The next step 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.
ax^2 + bx = -c
Step 3: Add and Subtract the Value to be Added
The value to be added is half of the coefficient of the x-term, squared. This value is added to both sides of the equation.
ax^2 + bx + (b/2)^2 = -c + (b/2)^2
Step 4: Factor the Left-Hand Side
The left-hand side of the equation can now be factored into the square of a binomial.
a(x + b/2)^2 = -c + (b/2)^2
Step 5: Simplify the Right-Hand Side
The right-hand side of the equation can be simplified by combining like terms.
a(x + b/2)^2 = (b/2)^2 - c
Step 6: Divide Both Sides by a
Finally, we divide both sides of the equation by a to isolate the binomial.
(x + b/2)^2 = ((b/2)^2 - c)/a
Example: Completing the Square
Let's consider an example to illustrate the process of completing the square.
x^2 + 6x + 8 = 0
Step 1: Identify the Quadratic Equation
The quadratic equation is x^2 + 6x + 8 = 0.
Step 2: Move the Constant Term to the Right-Hand Side
Subtracting 8 from both sides of the equation, we get:
x^2 + 6x = -8
Step 3: Add and Subtract the Value to be Added
Adding (6/2)^2 = 9 to both sides of the equation, we get:
x^2 + 6x + 9 = -8 + 9
Step 4: Factor the Left-Hand Side
The left-hand side of the equation can now be factored into the square of a binomial.
(x + 3)^2 = 1
Step 5: Simplify the Right-Hand Side
The right-hand side of the equation is already simplified.
(x + 3)^2 = 1
Step 6: Divide Both Sides by a
Since the coefficient of x^2 is 1, we can divide both sides of the equation by 1.
(x + 3)^2 = 1
Conclusion
Completing the square is a powerful algebraic technique used to solve quadratic equations and express them in a perfect square trinomial form. By following the steps outlined in this article, we can complete the square without using the formula. The example provided illustrates the process of completing the square, and we can see that the resulting equation is a perfect square trinomial.
In the expression, the value that was added to create the perfect square trinomial is being multiplied by -16, so the value that should be subtracted from 15 to maintain equivalency is ____
To solve this problem, we need to identify the value that was added to create the perfect square trinomial. In the example provided, the value added was 9. Since this value is being multiplied by -16, we need to multiply 9 by -16.
9 * -16 = -144
However, we need to subtract this value from 15 to maintain equivalency. Therefore, we need to subtract -144 from 15.
15 - (-144) = 15 + 144 = 159
Therefore, the value that should be subtracted from 15 to maintain equivalency is 159.
Table of Values
| Value | Description |
|---|---|
| 9 | Value added to create perfect square trinomial |
| -16 | Multiplier of value added |
| -144 | Result of multiplying value added by multiplier |
| 15 | Original value |
| 159 | Value to be subtracted from original value |
Conclusion
Introduction
Completing the square is a powerful algebraic technique used to solve quadratic equations and express them in a perfect square trinomial form. In our previous article, we provided a step-by-step guide on how to complete the square without using the formula. 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 process of transforming a quadratic equation into a perfect square trinomial form. This is achieved by adding and subtracting a constant term to the equation, which allows us to factor the equation into the square of a binomial.
Q: Why is completing the square useful?
A: Completing the square is useful because it allows us to solve quadratic equations in a more straightforward way. It also helps us to identify the vertex of a parabola, which is the point where the parabola changes direction.
Q: How do I know when to complete the square?
A: You should complete the square when you have a quadratic equation in the form of ax^2 + bx + c = 0, where a, b, and c are constants. Completing the square is particularly useful when the equation is not easily factorable.
Q: What is the value to be added?
A: The value to be added is half of the coefficient of the x-term, squared. This value is added to both sides of the equation.
Q: How do I find the value to be added?
A: To find the value to be added, you need to take half of the coefficient of the x-term and square it. For example, if the coefficient of the x-term is 6, you would take half of 6, which is 3, and square it, which gives you 9.
Q: What is the formula for completing the square?
A: The formula for completing the square is:
x^2 + bx + c = (x + b/2)^2 - (b/2)^2 + c
However, in our previous article, we provided a step-by-step guide on how to complete the square without using the formula.
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. However, you need to be careful when working with complex numbers, as they can be tricky to handle.
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. In fact, completing the square is often used to solve quadratic equations with rational coefficients.
Q: What are some common mistakes to avoid when completing the square?
A: Some common mistakes to avoid when completing the square include:
- Not adding and subtracting the value to be added correctly
- Not factoring the left-hand side of the equation correctly
- Not simplifying the right-hand side of the equation correctly
- Not checking the solutions to the equation
Q: How do I check the solutions to the equation?
A: To check the solutions to the equation, you need to plug the solutions back into the original equation and check if they are true. If the solutions are true, then they are valid solutions to the equation.
Conclusion
In this article, we have answered some frequently asked questions about completing the square. We have also provided some tips and tricks for completing the square, as well as some common mistakes to avoid. By following the steps outlined in our previous article and using the tips and tricks provided in this article, you should be able to complete the square with ease.
Additional Resources
- [Completing the Square: A Step-by-Step Guide](link to previous article)
- [Quadratic Equations: A Guide to Solving and Graphing](link to other article)
- [Algebra: A Comprehensive Guide](link to other article)
Practice Problems
- Complete the square for the following equation: x^2 + 4x + 5 = 0
- Solve the following quadratic equation using completing the square: x^2 + 2x - 3 = 0
- Find the value to be added for the following equation: x^2 + 6x + 9 = 0
Answer Key
- The value to be added is 4.
- The solutions to the equation are x = -1 and x = 3.
- The value to be added is 9.