Write $x^2 - 8x + 10$ In The Form $(x+a)^2 + B$.

Introduction

Quadratic expressions are a fundamental concept in algebra, and converting them into a specific form can be a challenging task. In this article, we will focus on converting the quadratic expression $x^2 - 8x + 10$ into the form $(x+a)^2 + b$. This process involves completing the square, which is a powerful technique for solving quadratic equations.

Understanding the Basics

Before we dive into the conversion process, let's review the basics of quadratic expressions. A quadratic expression is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Completing the Square

Completing the square is a technique used to convert a quadratic expression into a perfect square trinomial. This involves adding and subtracting a constant term to create a perfect square trinomial. The general form of a perfect square trinomial is $(x+a)^2$, where $a$ is a constant.

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 $-8$ in this case. We will use this coefficient to determine the value of $a$.

Step 2: Calculate the Value of $a$

To calculate the value of $a$, we need to divide the coefficient of the linear term by 2 and square the result. In this case, we have:

$$a = \frac{-8}{2} = -4$$

Step 3: Calculate the Value of $b$

To calculate the value of $b$, we need to square the value of $a$ and subtract the result from the constant term. In this case, we have:

$$b = 10 - (-4)^2 = 10 - 16 = -6$$

Step 4: Write the Quadratic Expression in the Desired Form

Now that we have calculated the values of $a$ and $b$, we can write the quadratic expression in the desired form:

$$(x-4)^2 - 6$$

Conclusion

Converting a quadratic expression into a specific form can be a challenging task, but with the right techniques and tools, it can be done. In this article, we have shown how to convert the quadratic expression $x^2 - 8x + 10$ into the form $(x+a)^2 + b$. By following the steps outlined in this article, you can convert any quadratic expression into the desired form.

Tips and Variations

• To convert a quadratic expression into the form $(x+a)^2 + b$, you need to identify the coefficient of the linear term and calculate the value of $a$.
• To calculate the value of $b$, you need to square the value of $a$ and subtract the result from the constant term.
• You can use the completing the square technique to solve quadratic equations.

Frequently Asked Questions

• Q: What is completing the square? A: Completing the square is a technique used to convert a quadratic expression into a perfect square trinomial.
• Q: How do I convert a quadratic expression into the form $(x+a)^2 + b$? A: To convert a quadratic expression into the form $(x+a)^2 + b$, you need to identify the coefficient of the linear term and calculate the value of $a$.
• Q: What is the value of $b$ in the quadratic expression $(x-4)^2 - 6$? A: The value of $b$ in the quadratic expression $(x-4)^2 - 6$ is $-6$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman

Glossary

• Quadratic expression: A polynomial of degree two.
• Completing the square: A technique used to convert a quadratic expression into a perfect square trinomial.
• Perfect square trinomial: A trinomial that can be written in the form $(x+a)^2$.
• Coefficient: A constant that is multiplied by a variable.
• Linear term: The term in a quadratic expression that has a highest power of one.
  Quadratic Expressions: A Q&A Guide =====================================

Introduction

Quadratic expressions are a fundamental concept in algebra, and understanding them can be a challenging task. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic expressions and how to work with them.

Q: What is a quadratic expression?

A: A quadratic expression is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the difference between a quadratic expression and a quadratic equation?

A: A quadratic expression is a polynomial of degree two, while a quadratic equation is an equation that contains a quadratic expression. For example, $x^2 + 4x + 4 = 0$ is a quadratic equation, while $x^2 + 4x + 4$ is a quadratic expression.

Q: How do I identify the coefficient of the linear term in a quadratic expression?

A: To identify the coefficient of the linear term, you need to look for the term that has a highest power of one. In the quadratic expression $x^2 + 4x + 4$, the coefficient of the linear term is $4$.

Q: What is completing the square?

A: Completing the square is a technique used to convert a quadratic expression into a perfect square trinomial. This involves adding and subtracting a constant term to create a perfect square trinomial.

Q: How do I complete the square?

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

1. Identify the coefficient of the linear term.
2. Calculate the value of $a$ by dividing the coefficient of the linear term by 2 and squaring the result.
3. Calculate the value of $b$ by squaring the value of $a$ and subtracting the result from the constant term.
4. Write the quadratic expression in the desired form.

Q: What is the value of $b$ in the quadratic expression $(x-4)^2 - 6$?

A: The value of $b$ in the quadratic expression $(x-4)^2 - 6$ is $-6$.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to follow these steps:

1. Write the quadratic equation in the form $ax^2 + bx + c = 0$.
2. Factor the quadratic expression, if possible.
3. Use the quadratic formula to find the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to find the solutions of a quadratic equation. The quadratic formula is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to follow these steps:

1. Write the quadratic equation in the form $ax^2 + bx + c = 0$.
2. Plug in the values of $a$, $b$, and $c$ into the quadratic formula.
3. Simplify the expression to find the solutions.

Q: What are the solutions of the quadratic equation $x^2 + 4x + 4 = 0$?

A: The solutions of the quadratic equation $x^2 + 4x + 4 = 0$ are $x = -2$ and $x = -2$.

Conclusion

Quadratic expressions are a fundamental concept in algebra, and understanding them can be a challenging task. In this article, we have provided a comprehensive Q&A guide to help you understand quadratic expressions and how to work with them. We hope that this guide has been helpful in answering your questions and providing you with a better understanding of quadratic expressions.

Tips and Variations

• To solve a quadratic equation, you need to follow the steps outlined in this article.
• To complete the square, you need to follow the steps outlined in this article.
• You can use the quadratic formula to find the solutions of a quadratic equation.

Frequently Asked Questions

• Q: What is a quadratic expression? A: A quadratic expression is a polynomial of degree two.
• Q: What is the difference between a quadratic expression and a quadratic equation? A: A quadratic expression is a polynomial of degree two, while a quadratic equation is an equation that contains a quadratic expression.
• Q: How do I identify the coefficient of the linear term in a quadratic expression? A: To identify the coefficient of the linear term, you need to look for the term that has a highest power of one.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman

Glossary

• Quadratic expression: A polynomial of degree two.
• Completing the square: A technique used to convert a quadratic expression into a perfect square trinomial.
• Perfect square trinomial: A trinomial that can be written in the form $(x+a)^2$.
• Coefficient: A constant that is multiplied by a variable.
• Linear term: The term in a quadratic expression that has a highest power of one.