Complete The Square To Write $y = 3x^2 + 12x + 7$ In Vertex Form, $y = A(x-h)^2 + K$.1. Start With: $y = 3(x^2 + 4x) + 7$2. Complete The Square Inside The Parentheses: $y = 3(x^2 + 4x + 4) + 7 - 3(4$\]

Introduction

In algebra, converting quadratic equations from standard form to vertex form is an essential skill that helps in understanding the properties of quadratic functions. The vertex form of a quadratic equation is given by $y = a(x-h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex of the parabola. In this article, we will focus on converting the quadratic equation $y = 3x^2 + 12x + 7$ to vertex form using the method of completing the square.

Step 1: Rewrite the Quadratic Equation

The first step in converting the quadratic equation to vertex form is to rewrite it in a form that allows us to complete the square. We start with the given equation:

$$y = 3x^2 + 12x + 7$$

To rewrite this equation, we can factor out the coefficient of the $x^2$ term, which is 3. This gives us:

$$y = 3(x^2 + 4x) + 7$$

Step 2: Complete the Square Inside the Parentheses

Now that we have the equation in the form $y = 3(x^2 + 4x) + 7$, we can complete the square inside the parentheses. To do this, we need to add and subtract the square of half the coefficient of the $x$ term inside the parentheses. The coefficient of the $x$ term is 4, so half of this is 2, and the square of 2 is 4.

We add 4 inside the parentheses and subtract 3 times 4 outside the parentheses to maintain the equation's equality:

$$y = 3(x^2 + 4x + 4) + 7 - 3(4)$$

Simplifying the Equation

Now that we have completed the square, we can simplify the equation by evaluating the expression inside the parentheses and combining like terms. The expression inside the parentheses is a perfect square trinomial, which can be written as:

$$(x + 2)^2$$

So, the equation becomes:

$$y = 3(x + 2)^2 + 7 - 12$$

Finalizing the Vertex Form

The final step is to simplify the constant term by evaluating the expression $7 - 12$. This gives us:

$$y = 3(x + 2)^2 - 5$$

Therefore, the quadratic equation $y = 3x^2 + 12x + 7$ can be written in vertex form as:

$$y = 3(x + 2)^2 - 5$$

Conclusion

In this article, we have demonstrated how to convert the quadratic equation $y = 3x^2 + 12x + 7$ to vertex form using the method of completing the square. By following the steps outlined above, we were able to rewrite the equation in the form $y = a(x-h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex of the parabola. This skill is essential in understanding the properties of quadratic functions and is a crucial tool in algebra and calculus.

Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is given by:

$$y = a(x-h)^2 + k$$

where $(h, k)$ represents the coordinates of the vertex of the parabola.

Properties of Quadratic Functions

Quadratic functions have several important properties, including:

• Vertex: The vertex of a quadratic function is the point at which the function reaches its maximum or minimum value.
• Axis of Symmetry: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the function.
• Intercepts: The x-intercepts of a quadratic function are the points at which the function crosses the x-axis, and the y-intercept is the point at which the function crosses the y-axis.

Real-World Applications of Quadratic Functions

Quadratic functions have numerous real-world applications, including:

• Projectile Motion: Quadratic functions are used to model the trajectory of projectiles, such as the flight of a baseball or the trajectory of a rocket.
• Optimization: Quadratic functions are used to optimize functions, such as finding the maximum or minimum value of a function.
• Physics: Quadratic functions are used to model the motion of objects, such as the motion of a pendulum or the motion of a spring.

Conclusion

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Introduction

Quadratic equations are a fundamental concept in algebra and calculus. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic equations and their applications.

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 (usually x) is two. The general form of a quadratic equation is:

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

where a, b, and c are constants, and a cannot be zero.

Q: What is the Vertex Form of a Quadratic Equation?

A: The vertex form of a quadratic equation is given by:

$$y = a(x-h)^2 + k$$

where (h, k) represents the coordinates of the vertex of the parabola.

Q: How Do I Convert a Quadratic Equation to Vertex Form?

A: To convert a quadratic equation to vertex form, you need to complete the square. Here's a step-by-step guide:

1. Factor out the coefficient of the x^2 term.
2. Add and subtract the square of half the coefficient of the x term inside the parentheses.
3. Simplify the equation by evaluating the expression inside the parentheses and combining like terms.

Q: What is the Axis of Symmetry of a Quadratic Equation?

A: The axis of symmetry of a quadratic equation is a vertical line that passes through the vertex of the parabola. It can be found using the formula:

$$x = -\frac{b}{2a}$$

Q: How Do I Find the X-Intercepts of a Quadratic Equation?

A: To find the x-intercepts of a quadratic equation, you need to set the equation equal to zero and solve for x. The x-intercepts are the points at which the function crosses the x-axis.

Q: What is the Y-Intercept of a Quadratic Equation?

A: The y-intercept of a quadratic equation is the point at which the function crosses the y-axis. It can be found by substituting x = 0 into the equation.

Q: What are the Real-World Applications of Quadratic Equations?

A: Quadratic equations have numerous real-world applications, including:

• Projectile Motion: Quadratic equations are used to model the trajectory of projectiles, such as the flight of a baseball 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.
• Physics: Quadratic equations are used to model the motion of objects, such as the motion of a pendulum or the motion of a spring.

Q: How Do I Solve a Quadratic Equation?

A: There are several methods to solve a quadratic equation, including:

• Factoring: If the quadratic equation can be factored, you can solve it by setting each factor equal to zero and solving for x.
• Quadratic Formula: The quadratic formula is a general method for solving quadratic equations. It is given by:

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

Conclusion

In conclusion, quadratic equations are a fundamental concept in algebra and calculus. By understanding the properties and applications of quadratic equations, you can solve a wide range of problems in mathematics and science. We hope this Q&A guide has been helpful in providing you with a comprehensive understanding of quadratic equations.

Additional Resources

For further learning, we recommend the following resources:

• Textbooks: "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by Michael Spivak.
• Online Resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
• Practice Problems: IXL, Mathway, and Symbolab.

Final Tips

• Practice, Practice, Practice: The best way to learn quadratic equations is by practicing problems.
• Understand the Concepts: Make sure you understand the concepts and properties of quadratic equations before moving on to more advanced topics.
• Use Online Resources: Online resources such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha can provide you with additional support and practice problems.