Express $x^2 + 4x - 7$ In The Form $(x + A)^2 - B$, Where $ A A A [/tex] And $b$ Are Integers.

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Introduction

In algebra, quadratic expressions are a fundamental concept that can be expressed in various forms. One of the most important forms is the vertex form, which is represented as $(x + a)^2 - b$. In this article, we will focus on expressing the quadratic expression $x^2 + 4x - 7$ in the form $(x + a)^2 - b$, where $a$ and $b$ are integers.

Understanding the Vertex Form

The vertex form of a quadratic expression is a way of expressing it in the form $(x + a)^2 - b$. This form is useful because it allows us to easily identify the vertex of the parabola represented by the quadratic expression. The vertex form is also useful for graphing quadratic functions.

Completing the Square

To express the quadratic expression $x^2 + 4x - 7$ in the form $(x + a)^2 - b$, we need to complete the square. Completing the square is a process of rewriting a quadratic expression in the form $(x + a)^2 - b$ by adding and subtracting a constant term.

Step 1: Identify the Coefficient of the x-term

The coefficient of the x-term in the quadratic expression $x^2 + 4x - 7$ is 4.

Step 2: Calculate the Constant Term to be Added

To complete the square, we need to add a constant term to the quadratic expression. The constant term to be added is calculated by taking half of the coefficient of the x-term and squaring it. In this case, half of 4 is 2, and squaring it gives us 4.

Step 3: Add and Subtract the Constant Term

We add and subtract the constant term 4 to the quadratic expression $x^2 + 4x - 7$ to get $x^2 + 4x + 4 - 4 - 7$.

Step 4: Rewrite the Quadratic Expression

We can now rewrite the quadratic expression $x^2 + 4x + 4 - 4 - 7$ as $(x + 2)^2 - 11$.

Conclusion

In this article, we have expressed the quadratic expression $x^2 + 4x - 7$ in the form $(x + a)^2 - b$, where $a$ and $b$ are integers. We have completed the square to rewrite the quadratic expression in the vertex form. The vertex form is a useful way of expressing quadratic expressions, and it allows us to easily identify the vertex of the parabola represented by the quadratic expression.

Example Problems

Problem 1

Express the quadratic expression $x^2 - 6x + 8$ in the form $(x + a)^2 - b$.

Solution

To express the quadratic expression $x^2 - 6x + 8$ in the form $(x + a)^2 - b$, we need to complete the square. We add and subtract a constant term to the quadratic expression to get $x^2 - 6x + 9 - 9 + 8$.

We can now rewrite the quadratic expression $x^2 - 6x + 9 - 9 + 8$ as $(x - 3)^2 - 1$.

Problem 2

Express the quadratic expression $x^2 + 2x - 3$ in the form $(x + a)^2 - b$.

Solution

To express the quadratic expression $x^2 + 2x - 3$ in the form $(x + a)^2 - b$, we need to complete the square. We add and subtract a constant term to the quadratic expression to get $x^2 + 2x + 1 - 1 - 3$.

We can now rewrite the quadratic expression $x^2 + 2x + 1 - 1 - 3$ as $(x + 1)^2 - 5$.

Applications of Vertex Form

The vertex form of a quadratic expression has many applications in mathematics and science. Some of the applications include:

  • Graphing Quadratic Functions: The vertex form is useful for graphing quadratic functions because it allows us to easily identify the vertex of the parabola.
  • Optimization Problems: The vertex form is useful for solving optimization problems because it allows us to easily identify the maximum or minimum value of a quadratic function.
  • Physics and Engineering: The vertex form is useful in physics and engineering because it allows us to easily model and analyze the motion of objects.

Conclusion

Introduction

In our previous article, we discussed how to express quadratic expressions in vertex form. In this article, we will answer some frequently asked questions about quadratic expressions in vertex form.

Q: What is the vertex form of a quadratic expression?

A: The vertex form of a quadratic expression is a way of expressing it in the form $(x + a)^2 - b$, where $a$ and $b$ are integers.

Q: How do I express a quadratic expression in vertex form?

A: To express a quadratic expression in vertex form, you need to complete the square. This involves adding and subtracting a constant term to the quadratic expression.

Q: What is completing the square?

A: Completing the square is a process of rewriting a quadratic expression in the form $(x + a)^2 - b$ by adding and subtracting a constant term.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

  1. Identify the coefficient of the x-term in the quadratic expression.
  2. Calculate the constant term to be added by taking half of the coefficient of the x-term and squaring it.
  3. Add and subtract the constant term to the quadratic expression.
  4. Rewrite the quadratic expression in the form $(x + a)^2 - b$.

Q: What are the applications of vertex form?

A: The vertex form of a quadratic expression has many applications in mathematics and science, including:

  • Graphing Quadratic Functions: The vertex form is useful for graphing quadratic functions because it allows us to easily identify the vertex of the parabola.
  • Optimization Problems: The vertex form is useful for solving optimization problems because it allows us to easily identify the maximum or minimum value of a quadratic function.
  • Physics and Engineering: The vertex form is useful in physics and engineering because it allows us to easily model and analyze the motion of objects.

Q: How do I use the vertex form to graph a quadratic function?

A: To use the vertex form to graph a quadratic function, you need to follow these steps:

  1. Identify the vertex of the parabola represented by the quadratic expression.
  2. Use the vertex to determine the direction of the parabola.
  3. Plot the vertex on the graph.
  4. Use the vertex form to determine the equation of the parabola.

Q: How do I use the vertex form to solve optimization problems?

A: To use the vertex form to solve optimization problems, you need to follow these steps:

  1. Identify the maximum or minimum value of the quadratic function.
  2. Use the vertex form to determine the equation of the parabola.
  3. Use the equation to determine the maximum or minimum value of the quadratic function.

Q: What are some common mistakes to avoid when working with vertex form?

A: Some common mistakes to avoid when working with vertex form include:

  • Not completing the square correctly: Make sure to complete the square correctly by following the steps outlined above.
  • Not identifying the vertex correctly: Make sure to identify the vertex correctly by using the vertex form.
  • Not using the vertex form correctly: Make sure to use the vertex form correctly by following the steps outlined above.

Conclusion

In conclusion, the vertex form of a quadratic expression is a useful way of expressing quadratic expressions. It allows us to easily identify the vertex of the parabola represented by the quadratic expression, and it has many applications in mathematics and science. We have answered some frequently asked questions about quadratic expressions in vertex form, and we hope that this article has been helpful in understanding the concept of vertex form.

Example Problems

Problem 1

Express the quadratic expression $x^2 - 6x + 8$ in the form $(x + a)^2 - b$.

Solution

To express the quadratic expression $x^2 - 6x + 8$ in the form $(x + a)^2 - b$, we need to complete the square. We add and subtract a constant term to the quadratic expression to get $x^2 - 6x + 9 - 9 + 8$.

We can now rewrite the quadratic expression $x^2 - 6x + 9 - 9 + 8$ as $(x - 3)^2 - 1$.

Problem 2

Express the quadratic expression $x^2 + 2x - 3$ in the form $(x + a)^2 - b$.

Solution

To express the quadratic expression $x^2 + 2x - 3$ in the form $(x + a)^2 - b$, we need to complete the square. We add and subtract a constant term to the quadratic expression to get $x^2 + 2x + 1 - 1 - 3$.

We can now rewrite the quadratic expression $x^2 + 2x + 1 - 1 - 3$ as $(x + 1)^2 - 5$.

Practice Exercises

Exercise 1

Express the quadratic expression $x^2 + 4x - 7$ in the form $(x + a)^2 - b$.

Exercise 2

Express the quadratic expression $x^2 - 2x - 3$ in the form $(x + a)^2 - b$.

Exercise 3

Use the vertex form to graph the quadratic function $y = x^2 + 4x - 7$.

Exercise 4

Use the vertex form to solve the optimization problem $\max x^2 + 4x - 7$.

Answer Key

Exercise 1

(x+2)2−11(x + 2)^2 - 11

Exercise 2

(x−1)2−6(x - 1)^2 - 6

Exercise 3

The graph of the quadratic function $y = x^2 + 4x - 7$ is a parabola with vertex $( - 2, - 11)$.

Exercise 4

The maximum value of the quadratic function $x^2 + 4x - 7$ is $- 11$, which occurs at $x = - 2$.