Simplify The Following Quadratic Equation And Rewrite It In Standard Form. After Rewriting, Identify The $a, B,$ And $c$ Terms.$g(x) = 3x^2 - 2(x - 1$\]Standard Form: $\square$ $a = \square$, $b =

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding how to simplify and rewrite them in standard form is crucial for solving various mathematical problems. In this article, we will focus on simplifying the given quadratic equation and rewriting it in standard form. We will also identify the coefficients of the quadratic equation, which are essential for further mathematical analysis.

The Given Quadratic Equation

The given quadratic equation is:

$$g(x) = 3x^2 - 2(x - 1)$$

Simplifying the Quadratic Equation

To simplify the quadratic equation, we need to apply the distributive property to the second term. The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. Applying this property to the second term, we get:

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

Rewriting the Quadratic Equation in Standard Form

The standard form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants. To rewrite the simplified quadratic equation in standard form, we need to rearrange the terms in the correct order. The simplified quadratic equation is already in the correct order, so we can rewrite it as:

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

Identifying the Coefficients

The coefficients of a quadratic equation are the constants that are multiplied by the variables. In the standard form of a quadratic equation, the coefficients are $a$, $b$, and $c$. To identify the coefficients, we need to look at the simplified quadratic equation:

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

• The coefficient of the $x^2$ term is $3$, so $a = 3$.
• The coefficient of the $x$ term is $-2$, so $b = -2$.
• The constant term is $2$, so $c = 2$.

Conclusion

In this article, we simplified the given quadratic equation and rewrote it in standard form. We also identified the coefficients of the quadratic equation, which are essential for further mathematical analysis. The simplified quadratic equation is:

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

The coefficients of the quadratic equation are:

• $a = 3$
• $b = -2$
• $c = 2$

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields, including physics, engineering, and economics. Some of the applications of quadratic equations include:

• Projectile Motion: Quadratic equations are used to model the trajectory of projectiles, such as the path of a thrown ball or the trajectory of a rocket.
• Optimization: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
• Data Analysis: Quadratic equations are used to model real-world data, such as the relationship between two variables.

Real-World Examples of Quadratic Equations

Quadratic equations have numerous real-world applications. Some examples include:

• Designing a Trajectory: Quadratic equations are used to design the trajectory of a spacecraft or a missile.
• Optimizing a Function: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
• Modeling Real-World Data: Quadratic equations are used to model real-world data, such as the relationship between two variables.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and understanding how to simplify and rewrite them in standard form is crucial for solving various mathematical problems. The simplified quadratic equation is:

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

The coefficients of the quadratic equation are:

• $a = 3$
• $b = -2$
• $c = 2$

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding how to simplify and rewrite them in standard form is crucial for solving various mathematical problems. In this article, we will provide a comprehensive Q&A section to help you better understand quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants.

Q: How do I simplify a quadratic equation?

A: To simplify a quadratic equation, you need to apply the distributive property to the second term. The distributive property states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. Applying this property to the second term, you can rewrite the quadratic equation in a simpler form.

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

A: The standard form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants.

Q: How do I identify the coefficients of a quadratic equation?

A: To identify the coefficients of a quadratic equation, you need to look at the simplified quadratic equation. The coefficients are:

• $a$: The coefficient of the $x^2$ term.
• $b$: The coefficient of the $x$ term.
• $c$: The constant term.

Q: What are the applications of quadratic equations?

A: Quadratic equations have numerous applications in various fields, including:

• Projectile Motion: Quadratic equations are used to model the trajectory of projectiles, such as the path of a thrown ball or the trajectory of a rocket.
• Optimization: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
• Data Analysis: Quadratic equations are used to model real-world data, such as the relationship between two variables.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use various methods, including:

• Factoring: If the quadratic equation can be factored, you can solve it by finding the roots of the equation.
• Quadratic Formula: If the quadratic equation cannot be factored, you can use the quadratic formula to solve it.

Q: What is the quadratic formula?

A: The quadratic formula is:

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

This formula can be used to solve quadratic equations that cannot be factored.

Q: What are the real-world examples of quadratic equations?

A: Quadratic equations have numerous real-world applications. Some examples include:

• Designing a Trajectory: Quadratic equations are used to design the trajectory of a spacecraft or a missile.
• Optimizing a Function: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
• Modeling Real-World Data: Quadratic equations are used to model real-world data, such as the relationship between two variables.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and understanding how to simplify and rewrite them in standard form is crucial for solving various mathematical problems. We hope this Q&A section has helped you better understand quadratic equations and their applications.