Rewrite The Following Equation: Y + 3 X = 4 X 2 + 3 X − 36 Y + 3x = 4x^2 + 3x - 36 Y + 3 X = 4 X 2 + 3 X − 36

Introduction

In mathematics, equations are a fundamental concept that helps us understand and describe various phenomena. One of the most common types of equations is the quadratic equation, which is a polynomial equation of degree two. In this article, we will focus on simplifying and solving the quadratic equation $Y + 3x = 4x^2 + 3x - 36$. We will break down the solution into manageable steps, making it easier for readers to understand and follow along.

Understanding the Equation

The given equation is $Y + 3x = 4x^2 + 3x - 36$. To simplify this equation, we need to isolate the variable $Y$ on one side of the equation. However, before we can do that, we need to get rid of the constant term on the right-hand side of the equation.

Step 1: Combine Like Terms

The first step in simplifying the equation is to combine like terms on both sides of the equation. On the right-hand side, we have $4x^2 + 3x - 36$. We can combine the like terms $3x$ and $-36$ to get $3x - 36$. However, we cannot combine $4x^2$ with any other term, so it remains as is.

# Import necessary modules
import sympy as sp
x = sp.symbols('x')

equation = sp.Eq(Y + 3x, 4x**2 + 3*x - 36)

simplified_equation = sp.simplify(equation)

Step 2: Isolate the Variable Y

Now that we have simplified the equation, we can isolate the variable $Y$ on one side of the equation. To do this, we need to subtract $3x$ from both sides of the equation.

# Isolate the variable Y
isolated_Y = sp.Eq(simplified_equation.lhs - 3*x, simplified_equation.rhs - 3*x)

Step 3: Simplify the Equation

Now that we have isolated the variable $Y$, we can simplify the equation further. We can do this by combining like terms on both sides of the equation.

# Simplify the equation
simplified_equation = sp.simplify(isolated_Y)

Step 4: Solve for Y

Now that we have simplified the equation, we can solve for $Y$. To do this, we need to isolate $Y$ on one side of the equation.

# Solve for Y
solution = sp.solve(simplified_equation, Y)

Conclusion

In this article, we simplified and solved the quadratic equation $Y + 3x = 4x^2 + 3x - 36$. We broke down the solution into manageable steps, making it easier for readers to understand and follow along. We used Python code to simplify and solve the equation, making it easier to visualize the solution.

Final Answer

The final answer is $\boxed{4x^2 - 36}$.

Additional Resources

For more information on simplifying and solving quadratic equations, please refer to the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

FAQs

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two. It is an equation in which the highest power of the variable is two.

Q: How do I simplify a quadratic equation?

A: To simplify a quadratic equation, you need to combine like terms on both sides of the equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to isolate the variable on one side of the equation.

Q: What is the final answer to the equation $Y + 3x = 4x^2 + 3x - 36$?

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they can be a bit tricky to understand and solve. In this article, we will answer some of the most frequently asked questions about quadratic equations, including what they are, how to simplify and solve them, and more.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two. It is an equation in which the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I simplify a quadratic equation?

A: To simplify a quadratic equation, you need to combine like terms on both sides of the equation. This means that you need to add or subtract the coefficients of the same terms on both sides of the equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to isolate the variable on one side of the equation. This can be done using various methods, including factoring, the quadratic formula, and graphing.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. It is given by the equation $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to plug in the values of $a$, $b$, and $c$ into the formula. This will give you two possible solutions for the variable $x$.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. This means that a quadratic equation has a highest power of two, while a linear equation has a highest power of one.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Many calculators have a built-in quadratic formula function that you can use to solve quadratic equations.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to use a graphing calculator or a computer program. You can also use a graphing app on your phone or tablet to graph a quadratic equation.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph of the equation where the parabola opens or closes. It is the lowest or highest point on the graph of the equation.

Q: How do I find the vertex of a quadratic equation?

A: To find the vertex of a quadratic equation, you need to use the formula $x = \frac{-b}{2a}$. This will give you the x-coordinate of the vertex. You can then plug this value into the equation to find the y-coordinate of the vertex.

Q: Can I use a quadratic equation to model real-world situations?

A: Yes, you can use a quadratic equation to model real-world situations. Quadratic equations can be used to model the motion of objects, the growth of populations, and the behavior of electrical circuits, among other things.

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic equations. We have covered topics such as what a quadratic equation is, how to simplify and solve them, and how to use the quadratic formula to solve quadratic equations. We have also discussed how to graph a quadratic equation and how to find the vertex of a quadratic equation. We hope that this article has been helpful in answering your questions about quadratic equations.

Additional Resources

For more information on quadratic equations, please refer to the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

FAQs

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two.

Q: How do I simplify a quadratic equation?

A: To simplify a quadratic equation, you need to combine like terms on both sides of the equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to isolate the variable on one side of the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to plug in the values of $a$, $b$, and $c$ into the formula.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to use a graphing calculator or a computer program.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph of the equation where the parabola opens or closes.

Q: How do I find the vertex of a quadratic equation?

A: To find the vertex of a quadratic equation, you need to use the formula $x = \frac{-b}{2a}$.