Rewrite The Equation In Standard Form: $ Y = -\frac{1}{4}(x-5)(x+3) $

Mar 13, 2025 by ADMIN 70 views

**Rewrite the Equation in Standard Form: A Comprehensive Guide**

Introduction

In mathematics, equations are a fundamental concept used to represent relationships between variables. Standard form is a crucial aspect of equation representation, as it provides a clear and concise way to express the relationship between variables. In this article, we will focus on rewriting the equation $ y = -\frac{1}{4}(x-5)(x+3) $ in standard form.

Understanding the Given Equation

The given equation is $ y = -\frac{1}{4}(x-5)(x+3) $. This equation represents a quadratic function, where the variable $ y $ is a function of the variable $ x $. The equation is in factored form, which means it is expressed as a product of two binomials.

Rewriting the Equation in Standard Form

To rewrite the equation in standard form, we need to expand the factored form and simplify the expression. The standard form of a quadratic equation is $ ax^2 + bx + c = 0 $, where $ a $, $ b $, and $ c $ are constants.

Expanding the Factored Form

To expand the factored form, we need to multiply the two binomials using the distributive property. The distributive property states that for any real numbers $ a $, $ b $, and $ c $, $ a(b+c) = ab + ac $.

import sympy as sp
x = sp.symbols('x')

equation = -1/4 * (x - 5) * (x + 3)

expanded_equation = sp.expand(equation)
print(expanded_equation)

The expanded equation is $ -\frac{1}{4}x^2 + \frac{5}{4}x + \frac{15}{4} $.

Simplifying the Expanded Form

To simplify the expanded form, we need to combine like terms. Like terms are terms that have the same variable and exponent.

import sympy as sp

x = sp.symbols('x')

equation = -1/4 * x**2 + 5/4 * x + 15/4

simplified_equation = sp.simplify(equation)
print(simplified_equation)

The simplified equation is $ -\frac{1}{4}x^2 + \frac{5}{4}x + \frac{15}{4} $.

Writing the Equation in Standard Form

To write the equation in standard form, we need to rewrite it in the form $ ax^2 + bx + c = 0 $.

import sympy as sp

x = sp.symbols('x')

equation = -1/4 * x**2 + 5/4 * x + 15/4

standard_form = sp.Eq(equation, 0)
print(standard_form)

The equation in standard form is $ -\frac{1}{4}x^2 + \frac{5}{4}x + \frac{15}{4} = 0 $.

Conclusion

In this article, we have rewritten the equation $ y = -\frac{1}{4}(x-5)(x+3) $ in standard form. We have used the distributive property to expand the factored form, combined like terms to simplify the expanded form, and rewritten the equation in the form $ ax^2 + bx + c = 0 $. The standard form of the equation is $ -\frac{1}{4}x^2 + \frac{5}{4}x + \frac{15}{4} = 0 $.

Applications of Standard Form

Standard form has numerous applications in mathematics and science. Some of the applications include:

Solving Quadratic Equations: Standard form is used to solve quadratic equations, which are equations of the form $ ax^2 + bx + c = 0 $.
Graphing Quadratic Functions: Standard form is used to graph quadratic functions, which are functions of the form $ f(x) = ax^2 + bx + c $.
Solving Systems of Equations: Standard form is used to solve systems of equations, which are sets of equations that involve multiple variables.

Real-World Examples

Standard form has numerous real-world applications. Some of the examples include:

Projectile Motion: Standard form is used to model the trajectory of a projectile, which is an object that is thrown or launched into the air.
Optimization Problems: Standard form is used to solve optimization problems, which are problems that involve finding the maximum or minimum value of a function.
Economics: Standard form is used to model economic systems, which are systems that involve the production, distribution, and consumption of goods and services.

Conclusion

Q: What is standard form?

A: Standard form is a way of representing an equation in the form $ ax^2 + bx + c = 0 $, where $ a $, $ b $, and $ c $ are constants.

Q: Why is standard form important?

A: Standard form is important because it provides a clear and concise way to express the relationship between variables. It is used to solve quadratic equations, graph quadratic functions, and solve systems of equations.

Q: How do I rewrite an equation in standard form?

A: To rewrite an equation in standard form, you need to expand the factored form using the distributive property, combine like terms, and rewrite the equation in the form $ ax^2 + bx + c = 0 $.

Q: What are some common mistakes to avoid when rewriting an equation in standard form?

A: Some common mistakes to avoid when rewriting an equation in standard form include:

Not expanding the factored form correctly
Not combining like terms correctly
Not rewriting the equation in the correct form

Q: How do I use standard form to solve quadratic equations?

A: To use standard form to solve quadratic equations, you need to set the equation equal to zero and then solve for the variable.

Q: How do I use standard form to graph quadratic functions?

A: To use standard form to graph quadratic functions, you need to identify the vertex of the parabola and then use the vertex to graph the function.

Q: What are some real-world applications of standard form?

A: Some real-world applications of standard form include:

Modeling the trajectory of a projectile
Solving optimization problems
Modeling economic systems

Q: How do I determine if an equation is in standard form?

A: To determine if an equation is in standard form, you need to check if it is in the form $ ax^2 + bx + c = 0 $, where $ a $, $ b $, and $ c $ are constants.

Q: Can I use standard form to solve systems of equations?

A: Yes, you can use standard form to solve systems of equations. To do this, you need to rewrite each equation in standard form and then solve the system of equations.

Q: What are some common challenges when working with standard form?

A: Some common challenges when working with standard form include:

Difficulty expanding the factored form
Difficulty combining like terms
Difficulty rewriting the equation in the correct form

Conclusion

In conclusion, standard form is a crucial aspect of equation representation, and it has numerous applications in mathematics and science. We have answered some frequently asked questions about standard form, and we have discussed some common challenges and real-world applications.