Which Equation Represents The Standard Form Of The Equation $y=(x+4)^2-3$?A. $y=x^2+8x+13$ B. \$y=x^2+4x-3$[/tex\] C. $y=x^2+4x+13$ D. $y=x^2+8x-3$

Introduction

In mathematics, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The standard form of a quadratic equation is a crucial concept in algebra, and it plays a vital role in solving quadratic equations. In this article, we will explore the standard form of a quadratic equation and provide a step-by-step guide on how to convert a given quadratic equation into its standard form.

What is the Standard Form of a Quadratic Equation?

The standard form of a quadratic equation is written in the form of $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The standard form of a quadratic equation is also known as the general form of a quadratic equation.

How to Convert a Quadratic Equation into its Standard Form?

To convert a quadratic equation into its standard form, we need to follow these steps:

1. Expand the squared term: If the quadratic equation is in the form of $y = (x + d)^2 + e$, we need to expand the squared term using the formula $(x + d)^2 = x^2 + 2dx + d^2$.
2. Simplify the equation: After expanding the squared term, we need to simplify the equation by combining like terms.
3. Write the equation in the standard form: Once we have simplified the equation, we can write it in the standard form of $y = ax^2 + bx + c$.

Example: Converting the Equation $y = (x + 4)^2 - 3$ into its Standard Form

Let's take the equation $y = (x + 4)^2 - 3$ as an example. To convert this equation into its standard form, we need to follow the steps mentioned above.

Step 1: Expand the Squared Term

We start by expanding the squared term using the formula $(x + d)^2 = x^2 + 2dx + d^2$. In this case, $d = 4$, so we have:

$$(x + 4)^2 = x^2 + 2(4)x + 4^2$$

Step 2: Simplify the Equation

Now, we simplify the equation by combining like terms:

$$x^2 + 2(4)x + 4^2 = x^2 + 8x + 16$$

Step 3: Write the Equation in the Standard Form

Finally, we write the equation in the standard form of $y = ax^2 + bx + c$:

$$y = x^2 + 8x + 16 - 3$$

$$y = x^2 + 8x + 13$$

Conclusion

In this article, we have discussed the standard form of a quadratic equation and provided a step-by-step guide on how to convert a given quadratic equation into its standard form. We have also used an example to illustrate the process of converting the equation $y = (x + 4)^2 - 3$ into its standard form. By following these steps, you can easily convert any quadratic equation into its standard form.

Which Equation Represents the Standard Form of the Equation $y = (x + 4)^2 - 3$?

Based on the steps mentioned above, we can conclude that the standard form of the equation $y = (x + 4)^2 - 3$ is:

$$y = x^2 + 8x + 13$$

Therefore, the correct answer is:

• A. $y = x^2 + 8x + 13$

Comparison of the Options

Let's compare the options given in the problem:

• A. $y = x^2 + 8x + 13$: This is the correct answer, as we have derived it in the previous section.
• B. $y = x^2 + 4x - 3$: This is not the correct answer, as it does not match the standard form of the equation $y = (x + 4)^2 - 3$.
• C. $y = x^2 + 4x + 13$: This is not the correct answer, as it does not match the standard form of the equation $y = (x + 4)^2 - 3$.
• D. $y = x^2 + 8x - 3$: This is not the correct answer, as it does not match the standard form of the equation $y = (x + 4)^2 - 3$.

Conclusion

$$$$

Q: What is the standard form of a quadratic equation?

A: The standard form of a quadratic equation is written in the form of $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I convert a quadratic equation into its standard form?

A: To convert a quadratic equation into its standard form, you need to follow these steps:

1. Expand the squared term: If the quadratic equation is in the form of $y = (x + d)^2 + e$, you need to expand the squared term using the formula $(x + d)^2 = x^2 + 2dx + d^2$.
2. Simplify the equation: After expanding the squared term, you need to simplify the equation by combining like terms.
3. Write the equation in the standard form: Once you have simplified the equation, you can write it in the standard form of $y = ax^2 + bx + c$.

Q: What is the difference between the standard form and the general form of a quadratic equation?

A: The standard form and the general form of a quadratic equation are the same. The standard form is written in the form of $y = ax^2 + bx + c$, while the general form is also written in the same form.

Q: Can I use the standard form to solve quadratic equations?

A: Yes, you can use the standard form to solve quadratic equations. Once you have the equation in the standard form, you can use various methods such as factoring, quadratic formula, or graphing to solve the equation.

Q: What are some common mistakes to avoid when converting a quadratic equation into its standard form?

A: Some common mistakes to avoid when converting a quadratic equation into its standard form include:

• Not expanding the squared term correctly: Make sure to use the correct formula to expand the squared term.
• Not simplifying the equation correctly: Make sure to combine like terms correctly.
• Not writing the equation in the standard form correctly: Make sure to write the equation in the standard form of $y = ax^2 + bx + c$.

Q: Can I use technology to help me convert a quadratic equation into its standard form?

A: Yes, you can use technology such as graphing calculators or computer algebra systems to help you convert a quadratic equation into its standard form.

Q: What are some real-world applications of the standard form of a quadratic equation?

A: The standard form of a quadratic equation has many real-world applications, including:

• Physics: The standard form is used to model the motion of objects under the influence of gravity or other forces.
• Engineering: The standard form is used to design and optimize systems such as bridges, buildings, and electronic circuits.
• Economics: The standard form is used to model economic systems and make predictions about future trends.

Conclusion

In conclusion, the standard form of a quadratic equation is a powerful tool that can be used to solve quadratic equations and model real-world phenomena. By following the steps outlined in this article, you can easily convert a quadratic equation into its standard form and use it to solve problems in various fields.