Using The Method Of Completing The Square, Rewrite $x^2 + 3x - 2$ In The Form $(x + B)^2 + C$.$x^2 + 3x - 2 =$Enter Your Next Step Here.

Feb 26, 2025 by ADMIN 137 views

**Completing the Square: A Method for Rewriting Quadratic Expressions**

Introduction to Completing the Square

Completing the square is a powerful method for rewriting quadratic expressions in a specific form, known as the vertex form. This form is particularly useful for understanding the properties of quadratic functions, such as their minimum or maximum values, and their corresponding x-coordinates. In this article, we will explore the method of completing the square and apply it to the quadratic expression $x^2 + 3x - 2$ .

The Method of Completing the Square

The method of completing the square involves creating a perfect square trinomial from a quadratic expression. This is done by adding and subtracting a specific value to the expression, which allows us to rewrite it in the desired form. The general form of a quadratic expression is $ax^2 + bx + c$ , and we want to rewrite it in the form $(x + b)^2 + c$ .

Step 1: Identify the Coefficient of the Linear Term

To complete the square, we need to identify the coefficient of the linear term, which is $b$ in the expression $ax^2 + bx + c$ . In our given expression, $x^2 + 3x - 2$ , the coefficient of the linear term is $3$ .

Step 2: Calculate the Value to be Added and Subtracted

To create a perfect square trinomial, we need to add and subtract a specific value to the expression. This value is calculated by taking half of the coefficient of the linear term and squaring it. In our case, half of $3$ is $1.5$ , and squaring it gives us $2.25$ . Therefore, we will add and subtract $2.25$ to the expression.

Step 3: Rewrite the Expression

Now that we have identified the value to be added and subtracted, we can rewrite the expression. We will add $2.25$ inside the parentheses and subtract $2.25$ outside the parentheses. This gives us:

$x^2 + 3x - 2 = (x^2 + 3x + 2.25) - 2.25 - 2$

Step 4: Simplify the Expression

We can simplify the expression by combining like terms. The expression inside the parentheses is a perfect square trinomial, which can be rewritten as $(x + 1.5)^2$ . Therefore, our expression becomes:

$x^2 + 3x - 2 = (x + 1.5)^2 - 4.25$

Conclusion

In this article, we have applied the method of completing the square to the quadratic expression $x^2 + 3x - 2$ . We have identified the coefficient of the linear term, calculated the value to be added and subtracted, rewritten the expression, and simplified it to obtain the desired form. The final expression is $(x + 1.5)^2 - 4.25$ , which is in the form $(x + b)^2 + c$ .

Example Applications of Completing the Square

Completing the square has many practical applications in mathematics and other fields. Some examples include:

Graphing Quadratic Functions: Completing the square allows us to rewrite quadratic expressions in the vertex form, which is particularly useful for graphing quadratic functions.
Solving Quadratic Equations: Completing the square can be used to solve quadratic equations by rewriting them in the form $(x + b)^2 = c$ .
Optimization Problems: Completing the square can be used to solve optimization problems by rewriting the objective function in the vertex form.

Common Mistakes to Avoid

When completing the square, there are several common mistakes to avoid. Some of these include:

Incorrectly Identifying the Coefficient of the Linear Term: Make sure to identify the correct coefficient of the linear term, as this will affect the value to be added and subtracted.
Incorrectly Calculating the Value to be Added and Subtracted: Make sure to calculate the correct value to be added and subtracted, as this will affect the final expression.
Not Simplifying the Expression: Make sure to simplify the expression by combining like terms, as this will affect the final form of the expression.

Conclusion

In conclusion, completing the square is a powerful method for rewriting quadratic expressions in the vertex form. By following the steps outlined in this article, we can rewrite the quadratic expression $x^2 + 3x - 2$ in the form $(x + b)^2 + c$ . This form is particularly useful for understanding the properties of quadratic functions and solving quadratic equations.

Introduction

Completing the square is a powerful method for rewriting quadratic expressions in the vertex form. In our previous article, we explored the method of completing the square and applied it to the quadratic expression $x^2 + 3x - 2$ . In this article, we will answer some common questions about completing the square and provide additional examples and explanations.

Q: What is the purpose of completing the square?

A: The purpose of completing the square is to rewrite a quadratic expression in the vertex form, which is particularly useful for understanding the properties of quadratic functions and solving quadratic equations.

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

A: You should use completing the square when you need to rewrite a quadratic expression in the vertex form, such as when graphing a quadratic function or solving a quadratic equation.

Q: What are the steps to complete the square?

A: The steps to complete the square are:

Identify the coefficient of the linear term.
Calculate the value to be added and subtracted.
Rewrite the expression by adding and subtracting the calculated value.
Simplify the expression by combining like terms.

Q: What is the value to be added and subtracted?

A: The value to be added and subtracted is calculated by taking half of the coefficient of the linear term and squaring it.

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

A: You can check if you have completed the square correctly by simplifying the expression and verifying that it is in the vertex form.

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

A: Yes, you can use completing the square to solve quadratic equations by rewriting the equation in the vertex form and then solving for the variable.

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

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

Incorrectly identifying the coefficient of the linear term.
Incorrectly calculating the value to be added and subtracted.
Not simplifying the expression by combining like terms.

Q: Can I use completing the square to graph quadratic functions?

A: Yes, you can use completing the square to graph quadratic functions by rewriting the function in the vertex form and then graphing the resulting parabola.

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

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

Graphing quadratic functions to model real-world situations.
Solving quadratic equations to find the maximum or minimum value of a function.
Optimizing problems by rewriting the objective function in the vertex form.

Q: Can I use completing the square to solve systems of equations?

A: Yes, you can use completing the square to solve systems of equations by rewriting the equations in the vertex form and then solving for the variables.

Conclusion

In conclusion, completing the square is a powerful method for rewriting quadratic expressions in the vertex form. By following the steps outlined in this article, you can answer common questions about completing the square and apply the method to solve quadratic equations and graph quadratic functions.

Additional Examples

Here are some additional examples of completing the square:

Example 1: Rewrite the quadratic expression $x^2 + 4x + 3$ in the vertex form.
Solution: The coefficient of the linear term is $4$ , so we calculate the value to be added and subtracted as $(4/2)^2 = 4$ . We then rewrite the expression as $(x^2 + 4x + 4) - 4 + 3 = (x + 2)^2 - 1$ .
Example 2: Solve the quadratic equation $x^2 + 2x - 3 = 0$ by completing the square.
Solution: We rewrite the equation as $(x^2 + 2x + 1) - 1 - 3 = 0$ , which simplifies to $(x + 1)^2 - 4 = 0$ . We then solve for the variable by setting the expression equal to zero and solving for $x$ .

Practice Problems

Here are some practice problems to help you apply the method of completing the square:

Problem 1: Rewrite the quadratic expression $x^2 + 5x + 2$ in the vertex form.
Problem 2: Solve the quadratic equation $x^2 + 3x - 2 = 0$ by completing the square.
Problem 3: Graph the quadratic function $f(x) = x^2 + 2x + 1$ by rewriting it in the vertex form.