Evaluate $-2x^2 + 5x - 3$ For $x = \frac{5}{4}$.

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Introduction

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Quadratic expressions are a fundamental concept in algebra, and evaluating them is a crucial skill for students to master. In this article, we will focus on evaluating the quadratic expression $-2x^2 + 5x - 3$ for a given value of $x$. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding Quadratic Expressions

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A quadratic expression is a polynomial expression of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In our example, the quadratic expression is $-2x^2 + 5x - 3$.

Evaluating the Quadratic Expression

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To evaluate the quadratic expression $-2x^2 + 5x - 3$ for $x = \frac{5}{4}$, we need to substitute the value of $x$ into the expression and simplify. Let's break down the process step by step.

Step 1: Substitute the Value of $x$

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The first step is to substitute the value of $x$ into the quadratic expression. We are given that $x = \frac{5}{4}$, so we will replace $x$ with $\frac{5}{4}$ in the expression.

import sympy as sp

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

# Define the quadratic expression
expr = -2*x**2 + 5*x - 3

# Substitute the value of x
x_value = 5/4
expr_substituted = expr.subs(x, x_value)

Step 2: Simplify the Expression

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After substituting the value of $x$, we need to simplify the expression. This involves combining like terms and performing any necessary arithmetic operations.

# Simplify the expression
expr_simplified = sp.simplify(expr_substituted)

Step 3: Evaluate the Expression

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The final step is to evaluate the simplified expression. This will give us the value of the quadratic expression for $x = \frac{5}{4}$.

# Evaluate the expression
result = expr_simplified

Conclusion

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Evaluating quadratic expressions is an essential skill for students to master. By following the steps outlined in this article, we can evaluate the quadratic expression $-2x^2 + 5x - 3$ for $x = \frac{5}{4}$. We used the sympy library in Python to perform the substitution and simplification steps. The final result is the value of the quadratic expression for the given value of $x$.

Example Use Case

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Evaluating quadratic expressions has numerous applications in various fields, including physics, engineering, and economics. For example, in physics, quadratic expressions are used to model the motion of objects under the influence of gravity. In engineering, quadratic expressions are used to design and optimize systems, such as bridges and buildings. In economics, quadratic expressions are used to model the behavior of markets and predict future trends.

Tips and Tricks

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• When evaluating quadratic expressions, make sure to substitute the value of $x$ correctly and simplify the expression carefully.
• Use a calculator or computer algebra system, such as sympy, to perform the substitution and simplification steps.
• Practice evaluating quadratic expressions regularly to build your skills and confidence.

Frequently Asked Questions

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• Q: What is a quadratic expression? A: A quadratic expression is a polynomial expression of degree two, which means it has a highest power of two.
• Q: How do I evaluate a quadratic expression? A: To evaluate a quadratic expression, substitute the value of $x$ into the expression and simplify.
• Q: What is the difference between a quadratic expression and a linear expression? A: A quadratic expression has a highest power of two, while a linear expression has a highest power of one.

Further Reading

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• For more information on quadratic expressions, see the Khan Academy video on quadratic expressions.
• For a comprehensive guide to algebra, see the book "Algebra" by Michael Artin.
• For a list of online resources for algebra, see the website "Math Open Reference".
  

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Introduction

---

Quadratic expressions are a fundamental concept in algebra, and evaluating them is a crucial skill for students to master. In this article, we will answer some of the most frequently asked questions about quadratic expressions.

Q&A

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Q: What is a quadratic expression?

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A: A quadratic expression is a polynomial expression of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I evaluate a quadratic expression?

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A: To evaluate a quadratic expression, substitute the value of $x$ into the expression and simplify. This involves combining like terms and performing any necessary arithmetic operations.

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

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A: A quadratic expression has a highest power of two, while a linear expression has a highest power of one. For example, the expression $2x^2 + 3x - 1$ is a quadratic expression, while the expression $2x + 3$ is a linear expression.

Q: How do I factor a quadratic expression?

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A: Factoring a quadratic expression involves expressing it as a product of two binomials. This can be done using the factoring method, which involves finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the quadratic formula?

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A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula can be used to find the solutions to a quadratic equation in the form $ax^2 + bx + c = 0$.

Q: How do I graph a quadratic expression?

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A: Graphing a quadratic expression involves plotting the points on a coordinate plane and drawing a smooth curve through them. This can be done using a graphing calculator or a computer algebra system.

Q: What is the vertex of a quadratic expression?

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A: The vertex of a quadratic expression is the point on the graph where the curve changes direction. It can be found using the formula:

$x = -\frac{b}{2a}$

Q: How do I find the x-intercepts of a quadratic expression?

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A: The x-intercepts of a quadratic expression are the points on the graph where the curve crosses the x-axis. They can be found by setting the expression equal to zero and solving for $x$.

Conclusion

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Quadratic expressions are a fundamental concept in algebra, and evaluating them is a crucial skill for students to master. By answering some of the most frequently asked questions about quadratic expressions, we hope to have provided a better understanding of this important topic.

Example Use Case

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Evaluating quadratic expressions has numerous applications in various fields, including physics, engineering, and economics. For example, in physics, quadratic expressions are used to model the motion of objects under the influence of gravity. In engineering, quadratic expressions are used to design and optimize systems, such as bridges and buildings. In economics, quadratic expressions are used to model the behavior of markets and predict future trends.

Tips and Tricks

---

• When evaluating quadratic expressions, make sure to substitute the value of $x$ correctly and simplify the expression carefully.
• Use a calculator or computer algebra system, such as sympy, to perform the substitution and simplification steps.
• Practice evaluating quadratic expressions regularly to build your skills and confidence.

Further Reading

---

• For more information on quadratic expressions, see the Khan Academy video on quadratic expressions.
• For a comprehensive guide to algebra, see the book "Algebra" by Michael Artin.
• For a list of online resources for algebra, see the website "Math Open Reference".