Solve For \[$ X \$\] In The Equation:$\[ 12x^2 + 13x = 5x \\]

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 quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. We will use the given equation $12x^2 + 13x = 5x$ as a case study to demonstrate the steps involved in solving quadratic equations.

Understanding the Equation

The given equation is $12x^2 + 13x = 5x$. To solve for $x$, we need to isolate the variable $x$ on one side of the equation. The first step is to simplify the equation by combining like terms.

Simplifying the Equation

To simplify the equation, we need to combine the like terms on the left-hand side of the equation.

# Import necessary modules
import sympy as sp

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

# Define the equation
equation = 12*x**2 + 13*x - 5*x

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

print(simplified_equation)

The simplified equation is $12x^2 + 8x = 0$.

Rearranging the Equation

The next step is to rearrange the equation so that all the terms are on one side of the equation.

# Rearrange the equation
rearranged_equation = 12*x**2 + 8*x

print(rearranged_equation)

The rearranged equation is $12x^2 + 8x = 0$.

Factoring the Equation

The next step is to factor the equation, if possible.

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

print(factored_equation)

The factored equation is $4x(3x + 2) = 0$.

Solving for x

The final step is to solve for $x$ by setting each factor equal to zero.

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

print(solution)

The solution is $x = 0$ or $x = -\frac{2}{3}$.

Conclusion

Solving quadratic equations is a crucial skill for students and professionals alike. In this article, we used the given equation $12x^2 + 13x = 5x$ as a case study to demonstrate the steps involved in solving quadratic equations. We simplified the equation, rearranged it, factored it, and finally solved for $x$. The solution is $x = 0$ or $x = -\frac{2}{3}$.

Tips and Tricks

• Always simplify the equation before solving it.
• Rearrange the equation so that all the terms are on one side of the equation.
• Factor the equation, if possible.
• Solve for $x$ by setting each factor equal to zero.

Common Mistakes

• Failing to simplify the equation before solving it.
• Not rearranging the equation so that all the terms are on one side of the equation.
• Not factoring the equation, if possible.
• Not solving for $x$ by setting each factor equal to zero.

Real-World Applications

Quadratic equations have numerous 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 bridges, buildings, and other structures.
• Economics: Quadratic equations are used to model the behavior of economic systems.
• Computer Science: Quadratic equations are used in computer graphics and game development.

Conclusion

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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 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 (usually x) 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 solve a quadratic equation?

A: To solve a quadratic equation, you need to follow these steps:

1. Simplify the equation by combining like terms.
2. Rearrange the equation so that all the terms are on one side of the equation.
3. Factor the equation, if possible.
4. Solve for x by setting each factor equal to zero.

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 (usually x) is one. It is typically written in the form ax + b = 0, where a and b are constants. Quadratic equations, on the other hand, have a degree of two, which means the highest power of the variable is 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 always a good idea to understand the steps involved in solving a quadratic equation, so you can verify the solution.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Failing to simplify the equation before solving it.
• Not rearranging the equation so that all the terms are on one side of the equation.
• Not factoring the equation, if possible.
• Not solving for x by setting each factor equal to zero.

Q: How do I determine if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you need to look at the discriminant (b^2 - 4ac). If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: Can I use quadratic equations to model real-world problems?

A: Yes, quadratic equations can be used to model real-world problems. Some examples include:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design bridges, buildings, and other structures.
• Economics: Quadratic equations are used to model the behavior of economic systems.
• Computer Science: Quadratic equations are used in computer graphics and game development.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to follow these steps:

1. Find the x-intercepts by setting the equation equal to zero and solving for x.
2. Find the y-intercept by substituting x = 0 into the equation.
3. Plot the x-intercepts and the y-intercept on a coordinate plane.
4. Draw a smooth curve through the points to form the graph of the 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 answered some of the most frequently asked questions about quadratic equations. We hope this article has been helpful in understanding quadratic equations and how to solve them.