Write The Quadratic Equation In Standard Form:$ -18 + 6x^2 - 7x = -5x + 2x^2 }$Answer { \square$ $

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Introduction

In mathematics, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. In this article, we will learn how to write a given quadratic equation in standard form.

Understanding the Given Equation

The given equation is -18 + 6x^2 - 7x = -5x + 2x^2. To write this equation in standard form, we need to simplify it by combining like terms.

Step 1: Simplify the Equation

First, let's simplify the equation by combining like terms. We can start by moving all the terms with x^2 to one side of the equation and the constant terms to the other side.

# Import necessary modules
import sympy as sp

x = sp.symbols('x')



equation = -18 + 6x**2 - 7x + 5x - 2x**2



simplified_equation = sp.simplify(equation)

Step 2: Write the Equation in Standard Form

Now that we have simplified the equation, we can write it in standard form by rearranging the terms.

# Rearrange the terms to write the equation in standard form
standard_form = sp.simplify(simplified_equation)

Solution

The simplified equation is 4x^2 - 2x - 18. To write this equation in standard form, we can rearrange the terms as follows:

4x^2 - 2x - 18 = 0

This is the standard form of the quadratic equation.

Conclusion

In this article, we learned how to write a given quadratic equation in standard form. We simplified the equation by combining like terms and then rearranged the terms to write it in standard form. The standard form of the quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. We can use this form to solve quadratic equations and find the solutions.

Example Use Cases

Quadratic equations have many real-world applications, such as:

  • Projectile Motion: The trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation.
  • Optimization: Quadratic equations can be used to find the maximum or minimum value of a function.
  • Electrical Engineering: Quadratic equations are used to analyze and design electrical circuits.

Tips and Tricks

  • Use the Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It is given by x = (-b ± √(b^2 - 4ac)) / 2a.
  • Check for Extraneous Solutions: When solving quadratic equations, it is essential to check for extraneous solutions. An extraneous solution is a solution that does not satisfy the original equation.

Final Thoughts

Frequently Asked Questions

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. The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.

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, we can set each factor equal to zero and solve for x.
  • Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It is given by x = (-b ± √(b^2 - 4ac)) / 2a.
  • Graphing: We can graph the quadratic equation and find the x-intercepts to solve for x.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by x = (-b ± √(b^2 - 4ac)) / 2a.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, we need to plug in the values of a, b, and c into the formula. Then, we can simplify the expression and solve for x.

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. A quadratic equation, on the other hand, is a polynomial equation of degree two, which means the highest power of the variable is two.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula only gives two possible values for x.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions. This occurs when the discriminant (b^2 - 4ac) is negative.

Q: What is the discriminant?

A: The discriminant is the expression b^2 - 4ac in the quadratic formula. It determines the nature of the solutions of the quadratic equation.

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

A: To determine the nature of the solutions of a quadratic equation, we need to examine the discriminant. If the discriminant is:

  • Positive: The quadratic equation has two distinct real solutions.
  • Zero: The quadratic equation has one real solution.
  • Negative: The quadratic equation has no real solutions.

Q: Can a quadratic equation have complex solutions?

A: Yes, a quadratic equation can have complex solutions. This occurs when the discriminant is negative.

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

A: To find the complex solutions of a quadratic equation, we need to use the quadratic formula and take the square root of the negative discriminant.

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic equations. We have discussed the definition of a quadratic equation, how to solve a quadratic equation, and the nature of the solutions of a quadratic equation. We have also discussed the quadratic formula and how to use it to solve quadratic equations.