Problem 67 - Algebra 2

Question

Determine whether the given value satisfies the inequality. $$ -2 x^{2}+3 < 0 ; x=5 $$

Step-by-Step Solution

Answer

The value satisfies the inequality.

Step 1: Substitute the value of x

First, substitute the given value \( x = 5 \) into the inequality \( -2x^2 + 3 < 0 \).

Step 2: Calculate the expression

Evaluate \( -2(5)^2 + 3 \). Calculate \( (5)^2 = 25 \), then \( -2\times 25 = -50 \). Finally, compute \( -50 + 3 = -47 \).

Step 3: Compare with zero

Now that we have \( -47 \), compare this to zero to determine if \( -47 < 0 \).

Step 4: Conclusion

Since \( -47 < 0 \) is true, we conclude that the given value satisfies the inequality.

Key Concepts

Understanding Quadratic Expressions

Quadratic expressions are a fundamental concept in mathematics, often forming the backbone of many algebraic problems. A quadratic expression is an expression that can be written in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). Quadratic expressions are prevalent because they allow us to model a wide range of real-world scenarios such as projectile paths, profit optimization, and much more. In the expression \(-2x^2 + 3\), which is part of our problem, the term \(-2x^2\) represents the quadratic component. This component determines the parabola's opening direction and width based on the sign and magnitude of \( a \), which in this case is \(-2\). Understanding how to manipulate and work with quadratic expressions is vital as they are foundational to solving equations and inequalities involving squares.

The Substitution Method

The substitution method is a simple yet powerful tool used to solve equations and inequality problems like the one given. The essence of this method lies in replacing a variable with a given number. For instance, in our example, we substitute \( x = 5 \) into the inequality \( -2x^2 + 3 < 0 \). This step involves replacing \( x \) with 5 in each occurrence in the expression, which helps to transform the algebraic expression into a simple arithmetic problem.

• First, calculate \( (5)^2 = 25 \).
• Then perform the multiplication: \(-2 \times 25 = -50\).
• Finally, adjust by adding 3, resulting in \(-47\).

Substitution is invaluable when checking if a specific value satisfies a given equation or inequality.

Solving Inequality Solutions

Inequalities are mathematical expressions involving the symbols \( <, >, \leq, \geq \), and learning how to solve them is crucial. Solving inequalities often involves determining whether a statement is true or false under given conditions. In our original exercise, we evaluate whether \( -47 < 0 \). In simpler terms, we check if the obtained result from substituting \( x \) and simplifying the expression holds true within the confines of the inequality symbol.

• Evaluate the expression after substitution.
• Compare the result with zero or another quantity as specified by the inequality.
• If the comparison holds true, the inequality is satisfied for that value.

Mastering inequality solutions can extend to more complex scenarios, increasing proficiency in problem-solving and critical thinking.

Properties of Polynomials

Polynomials, including quadratics, are expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding the nature of polynomials is fundamental in algebra. For a given polynomial like \(-2x^2 + 3\), different values of \( x \) will yield different results, affecting whether an equation or inequality holds true. The structure of a polynomial is key:

• The leading term (highest power), \(-2x^2\), dictates the polynomial's overall shape.
• The constant term \(+3\) shifts the polynomial up or down on a graph.
• The interplay between the polynomial's terms affects evaluations on calculations, impacting how solutions like inequalities behave for various \( x \) values.

Through understanding polynomials, students gain the ability to dynamically approach problems, allowing for adaptability in both academic and real world contexts.