Solve By Completing The Square:${ F(x) = -0.04x^2 - 1.32x + 3.11 }$

Mar 11, 2025 by ADMIN 69 views

Solve by Completing the Square: A Step-by-Step Guide to Quadratic Equations

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Introduction

Completing the square is a powerful technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can be easily solved. In this article, we will explore the concept of completing the square and provide a step-by-step guide on how to solve quadratic equations using this method.

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 goal is to rewrite the equation in the form (x + d)^2 = e, where d and e are constants. This form can be easily solved by taking the square root of both sides.

The Process of Completing the Square

The process of completing the square involves the following steps:

Write the equation in the standard form: The equation should be in the form ax^2 + bx + c = 0.
Move the constant term to the right-hand side: This will isolate the terms involving x on the left-hand side.
Divide the coefficient of x^2 by 2: This will help us find the value to add to both sides of the equation.
Square the value found in step 3: This will give us the value to add to both sides of the equation.
Add the value found in step 4 to both sides of the equation: This will complete the square.
Simplify the equation: This will give us the final form of the equation.

Step-by-Step Guide to Solving Quadratic Equations by Completing the Square

Let's use the given equation f(x) = -0.04x^2 - 1.32x + 3.11 as an example.

Step 1: Write the Equation in the Standard Form

The equation is already in the standard form: f(x) = -0.04x^2 - 1.32x + 3.11.

Step 2: Move the Constant Term to the Right-Hand Side

f(x) = -0.04x^2 - 1.32x

Step 3: Divide the Coefficient of x^2 by 2

a = -0.04 d = a/2 = -0.02

Step 4: Square the Value Found in Step 3

d^2 = (-0.02)^2 = 0.0004

Step 5: Add the Value Found in Step 4 to Both Sides of the Equation

f(x) = -0.04x^2 - 1.32x + 0.0004

Step 6: Simplify the Equation

f(x) = (-0.02x - 0.66)^2 + 3.11 - 0.0004

f(x) = (-0.02x - 0.66)^2 + 3.1096

Conclusion

Completing the square is a powerful technique used to solve quadratic equations. By following the steps outlined in this article, you can easily solve quadratic equations using this method. Remember to write the equation in the standard form, move the constant term to the right-hand side, divide the coefficient of x^2 by 2, square the value found, add the value to both sides of the equation, and simplify the equation.

Example Problems

Problem 1

Solve the equation f(x) = 2x^2 + 3x - 1 using the completing the square method.

Solution

f(x) = 2x^2 + 3x - 1 f(x) = 2x^2 + 3x f(x) = 2(x^2 + 1.5x) f(x) = 2(x^2 + 1.5x + 0.75) - 2(0.75) f(x) = 2(x + 0.75)^2 - 1.5 f(x) = 2(x + 0.75)^2 - 1.5

Problem 2

Solve the equation f(x) = -x^2 + 2x + 3 using the completing the square method.

Solution

f(x) = -x^2 + 2x + 3 f(x) = -x^2 + 2x f(x) = -(x^2 - 2x) f(x) = -(x^2 - 2x + 1) + 1 f(x) = -(x - 1)^2 + 1 f(x) = -(x - 1)^2 + 1

Tips and Tricks

Make sure to write the equation in the standard form before starting the completing the square method.
Divide the coefficient of x^2 by 2 to find the value to add to both sides of the equation.
Square the value found in step 3 to get the value to add to both sides of the equation.
Add the value found in step 4 to both sides of the equation to complete the square.
Simplify the equation to get the final form.

Common Mistakes to Avoid

Not writing the equation in the standard form before starting the completing the square method.
Not dividing the coefficient of x^2 by 2 to find the value to add to both sides of the equation.
Not squaring the value found in step 3 to get the value to add to both sides of the equation.
Not adding the value found in step 4 to both sides of the equation to complete the square.
Not simplifying the equation to get the final form.

Conclusion

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Introduction

Completing the square is a powerful technique used to solve quadratic equations. However, it can be a bit tricky to understand and apply, especially for those who are new to it. In this article, we will answer some of the most frequently asked questions about completing the square, providing you with a better understanding of this technique and how to use it to solve quadratic equations.

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. It involves manipulating the equation to express it in a perfect square form, which can be easily solved.

Q: How do I know if I should use completing the square to solve a quadratic equation?

A: You should use completing the square to solve a quadratic equation when the equation is not easily factorable and you need to find the solutions. Completing the square is a good option when the equation has a negative coefficient of x^2, as it can help you to find the solutions more easily.

Q: What are the steps involved in completing the square?

A: The steps involved in completing the square are:

Write the equation in the standard form: The equation should be in the form ax^2 + bx + c = 0.
Move the constant term to the right-hand side: This will isolate the terms involving x on the left-hand side.
Divide the coefficient of x^2 by 2: This will help you to find the value to add to both sides of the equation.
Square the value found in step 3: This will give you the value to add to both sides of the equation.
Add the value found in step 4 to both sides of the equation: This will complete the square.
Simplify the equation: This will give you the final form of the equation.

Q: How do I know if I have completed the square correctly?

A: You can check if you have completed the square correctly by looking at the equation. If the equation is in the form (x + d)^2 = e, where d and e are constants, then you have completed the square correctly.

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

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

Not writing the equation in the standard form before starting the completing the square method.
Not dividing the coefficient of x^2 by 2 to find the value to add to both sides of the equation.
Not squaring the value found in step 3 to get the value to add to both sides of the equation.
Not adding the value found in step 4 to both sides of the equation to complete the square.
Not simplifying the equation to get the final form.

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 will need to use complex numbers and follow the same steps as before.

Q: How do I use completing the square to solve quadratic equations with rational coefficients?

A: You can use completing the square to solve quadratic equations with rational coefficients by following the same steps as before. However, you will need to simplify the equation to get the final form.

Q: Can I use completing the square to solve quadratic equations with irrational coefficients?

A: Yes, you can use completing the square to solve quadratic equations with irrational coefficients. However, you will need to use the properties of irrational numbers and follow the same steps as before.

Q: How do I know if a quadratic equation can be solved using completing the square?

A: You can check if a quadratic equation can be solved using completing the square by looking at the equation. If the equation is in the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to 0, then you can use completing the square to solve the equation.

Q: What are some real-world applications of completing the square?

A: Some real-world applications of completing the square include:

Physics: Completing the square is used to solve equations of motion and to find the position and velocity of an object.
Engineering: Completing the square is used to solve equations of electrical circuits and to find the current and voltage of a circuit.
Computer Science: Completing the square is used to solve equations of algorithms and to find the time and space complexity of an algorithm.