Solve The Equation:$\[ 9x^2 - 10x = X^2 + 4x \\]

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the equation 9x2−10x=x2+4x9x^2 - 10x = x^2 + 4x. We will break down the solution into manageable steps, using algebraic techniques to isolate the variable and find the solutions.

Understanding the Equation

The given equation is a quadratic equation, which can be written in the standard form as ax2+bx+c=0ax^2 + bx + c = 0. In this case, the equation is 9x2−10x=x2+4x9x^2 - 10x = x^2 + 4x. To solve this equation, we need to first simplify it by combining like terms.

Simplifying the Equation

To simplify the equation, we can start by moving all the terms to one side of the equation. This will give us a quadratic equation in the standard form.

# Import necessary modules
import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the equation
equation = 9*x**2 - 10*x - (x**2 + 4*x)

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

The simplified equation is 8x2−14x=08x^2 - 14x = 0. Now, we can factor out the common term 2x2x from the equation.

Factoring the Equation

Factoring the equation will help us isolate the variable and find the solutions.

# Factor the equation
factored_equation = sp.factor(simplified_equation)

The factored equation is 2x(4x−7)=02x(4x - 7) = 0. Now, we can set each factor equal to zero and solve for the variable.

Solving for the Variable

To solve for the variable, we can set each factor equal to zero and solve for xx.

# Solve for the variable
solution = sp.solve(factored_equation, x)

The solution to the equation is x=0x = 0 or x=74x = \frac{7}{4}.

Conclusion

Solving quadratic equations is a crucial skill for students and professionals alike. In this article, we focused on solving the equation 9x2−10x=x2+4x9x^2 - 10x = x^2 + 4x. We broke down the solution into manageable steps, using algebraic techniques to isolate the variable and find the solutions. By following these steps, we can solve quadratic equations and find the solutions.

Tips and Tricks

  • When solving quadratic equations, it's essential to simplify the equation by combining like terms.
  • Factoring the equation can help us isolate the variable and find the solutions.
  • Setting each factor equal to zero and solving for the variable is a crucial step in solving quadratic equations.

Common Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and there are many types of quadratic equations. Some common quadratic equations include:

  • ax2+bx+c=0ax^2 + bx + c = 0
  • x2+bx+c=0x^2 + bx + c = 0
  • ax2−bx+c=0ax^2 - bx + c = 0

Real-World Applications

Quadratic equations have many real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.

Conclusion

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will answer some of the most frequently asked questions about 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 can be written in the standard form as ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

  1. Simplify the equation by combining like terms.
  2. Factor the equation, if possible.
  3. Set each factor equal to zero and solve for the variable.

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

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable is one. It can be written in the standard form as ax+b=0ax + b = 0, where aa and bb are constants. A quadratic equation, on the other hand, is a polynomial equation of degree two.

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

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in function to solve quadratic equations. However, it's essential to understand the steps involved in solving a quadratic equation, as this will help you to check your work and understand the solution.

Q: What is the discriminant in a quadratic equation?

A: The discriminant is a value that can be calculated from the coefficients of a quadratic equation. It is used to determine the nature of the solutions to the equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: How do I determine the nature of the solutions to a quadratic equation?

A: To determine the nature of the solutions to a quadratic equation, you can use the discriminant. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

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

A: Yes, you can use the quadratic formula to solve a quadratic equation. The quadratic formula is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula can be used to find the solutions to a quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to find the solutions to a quadratic equation. It is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula can be used to find the solutions to a quadratic equation.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have answered some of the most frequently asked questions about quadratic equations. By understanding the steps involved in solving a quadratic equation and using the quadratic formula, you can solve quadratic equations and find the solutions.

Tips and Tricks

  • When solving a quadratic equation, it's essential to simplify the equation by combining like terms.
  • Factoring the equation can help you to isolate the variable and find the solutions.
  • The quadratic formula can be used to find the solutions to a quadratic equation.

Common Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and there are many types of quadratic equations. Some common quadratic equations include:

  • ax2+bx+c=0ax^2 + bx + c = 0
  • x2+bx+c=0x^2 + bx + c = 0
  • ax2−bx+c=0ax^2 - bx + c = 0

Real-World Applications

Quadratic equations have many real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.