Simplifying Quadratic Equations

Oct 13, 2025 by wbrum.org 98 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will explore the concept of quadratic equations and learn how to simplify and write them in standard form. We will also examine a specific polynomial and determine which quadratic function is equivalent to it.

What are Quadratic Equations?

A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (usually x or c) 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.

Simplifying Quadratic Equations

To simplify a quadratic equation, we need to combine like terms and rewrite the equation in standard form. The standard form of a quadratic equation is:

ax^2 + bx + c = 0

where a, b, and c are constants, and x is the variable.

Writing Quadratic Equations in Standard Form

To write a quadratic equation in standard form, we need to follow these steps:

Combine like terms: Combine the terms with the same variable and exponent.
Move the constant term to the right-hand side: Move the constant term to the right-hand side of the equation.
Write the equation in the form ax^2 + bx + c = 0: Write the equation in the standard form.

Example: Simplifying a Quadratic Equation

Let's consider the quadratic equation:

2x^2 + 5x - 3 = 0

To simplify this equation, we need to combine like terms and rewrite it in standard form.

2x^2 + 5x - 3 = 0

Combine like terms:

2x^2 + 5x = 3

Move the constant term to the right-hand side:

2x^2 + 5x - 3 = 0

Write the equation in standard form:

2x^2 + 5x - 3 = 0

The Original Polynomial

The original polynomial is:

c^2 - 4c - 7

We need to simplify this polynomial and write it in standard form.

Simplifying the Original Polynomial

To simplify the original polynomial, we need to combine like terms and rewrite it in standard form.

c^2 - 4c - 7

Combine like terms:

c^2 - 4c = -7

Move the constant term to the right-hand side:

c^2 - 4c + 7 = 0

Write the equation in standard form:

c^2 - 4c + 7 = 0

Which Quadratic Function is Equivalent to the Polynomial?

We have two quadratic functions:

A. -4c^2 + 4c + 6 B. -4c^2 + 4c + 8 C. -7c^2 + 7c + 6 D. -7c^2 + 10c + 6

We need to determine which quadratic function is equivalent to the polynomial c^2 - 4c - 7.

Comparing the Quadratic Functions

Let's compare the quadratic functions with the polynomial:

A. -4c^2 + 4c + 6

This quadratic function has a negative coefficient for c^2, which is not present in the polynomial.

B. -4c^2 + 4c + 8

This quadratic function has a negative coefficient for c^2, which is not present in the polynomial.

C. -7c^2 + 7c + 6

This quadratic function has a negative coefficient for c^2, which is not present in the polynomial.

D. -7c^2 + 10c + 6

This quadratic function has a negative coefficient for c^2, which is not present in the polynomial.

However, if we multiply the polynomial by -1, we get:

c^2 + 4c + 7

This is equivalent to the quadratic function:

-7c^2 - 7c - 7

But we can rewrite this as:

-7c^2 + 7c + 6

This is the same as option C.

Conclusion

In conclusion, the quadratic function that is equivalent to the polynomial c^2 - 4c - 7 is:

C. -7c^2 + 7c + 6

This quadratic function has the same coefficients as the polynomial, but with a negative coefficient for c^2.

Final Answer

Introduction

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (usually x or c) 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 and rewrite the equation in standard form. The standard form of a quadratic equation is:

ax^2 + bx + c = 0

where a, b, and c are constants, and x is the variable.

Q: What is the standard form of a quadratic equation?

A: The standard 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 write a quadratic equation in standard form?

A: To write a quadratic equation in standard form, you need to follow these steps:

Combine like terms: Combine the terms with the same variable and exponent.
Move the constant term to the right-hand side: Move the constant term to the right-hand side of the equation.
Write the equation in the form ax^2 + bx + c = 0: Write the equation in the standard form.

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. The general form of a linear equation is:

ax + b = 0

where a and b are constants, and x is the variable.

Q: Can I solve a quadratic equation by factoring?

A: Yes, you can solve a quadratic equation by factoring. However, not all quadratic equations can be factored. If the equation cannot be factored, you may need to use other methods such as the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a method for solving quadratic equations that cannot be factored. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are constants, and x is the variable.

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

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex roots. However, the roots will be in the form of complex numbers.

Q: What is the difference between a quadratic equation with real roots and a quadratic equation with complex roots?

A: A quadratic equation with real roots has roots that are real numbers, while a quadratic equation with complex roots has roots that are complex numbers.

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

A: Yes, you can use the quadratic formula to solve a quadratic equation with rational roots. However, the roots will be in the form of rational numbers.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. We have answered some frequently asked questions about quadratic equations, including how to simplify a quadratic equation, the standard form of a quadratic equation, and how to use the quadratic formula to solve a quadratic equation.

Final Answer

The final answer is that quadratic equations are a powerful tool for solving problems in mathematics and other fields. By understanding how to simplify a quadratic equation, write it in standard form, and use the quadratic formula, you can solve a wide range of problems.