Problem 79 - Algebra 2

Question

Evaluate \(b^{2}-4 a c\) for the given values of \(a, b,\) and \(c\). \(a=1, b=2, c=5\)

Step-by-Step Solution

Answer

The result is -16.

Step 1: Substitute the Values

Substitute the values of \(a\), \(b\), and \(c\) into the expression \(b^{2} - 4ac\). Given \(a = 1\), \(b = 2\), and \(c = 5\), the expression becomes \(2^{2} - 4 \times 1 \times 5\).

Step 2: Calculate \(b^2\)

Calculate \(b^2\) by squaring \(b\). Since \(b = 2\), \(b^2 = 2^2 = 4\).

Step 3: Calculate \(4ac\)

Calculate \(4ac\) by multiplying 4, \(a\), and \(c\). Since \(a = 1\) and \(c = 5\), \(4ac = 4 \times 1 \times 5 = 20\).

Step 4: Subtract \(4ac\) from \(b^2\)

Subtract \(4ac\) from \(b^2\) to get the result. Thus, \(b^{2} - 4ac = 4 - 20 = -16\).

Key Concepts

Quadratic Formula

The quadratic formula is a cornerstone in solving quadratic equations. These equations are typically written in the form \(ax^2 + bx + c = 0\). The quadratic formula provides a way to find the values of \(x\) corresponding to any given set of coefficients \(a\), \(b\), and \(c\). The formula is as follows:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, the expression under the square root sign, \(b^2 - 4ac\), is called the discriminant. The discriminant is crucial because it helps determine the nature of the roots (solutions) of the quadratic equation.

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (also known as a repeated root).
• If \(b^2 - 4ac < 0\), the roots are complex and occur in conjugate pairs.

Understanding the discriminant is key to solving and predicting the behavior of quadratic equations effectively.

Evaluate Expression

Evaluating an expression simply means calculating its value using defined operations and substitutions. In the context of algebra, this often involves substituting values into variables and performing arithmetic operations. For the expression \(b^2 - 4ac\) given specific values of \(a\), \(b\), and \(c\), evaluation requires us to:

• Start with the formula \(b^2 - 4ac\).
• Substitute the given values into the expression. For instance, with \(a = 1\), \(b = 2\), and \(c = 5\), the expression becomes \(2^2 - 4 \times 1 \times 5\).
• Perform the calculations in sequence: first calculate \(b^2\), then \(4ac\).
• Subtract \(4ac\) from \(b^2\) to find the final result.

Each step helps simplify the expression until you reach the numerical result, ensuring clarity and precision throughout the process. This process is foundational in algebra, leading to a thorough understanding of how expressions behave when variables are altered.

Algebraic Substitution

Algebraic substitution is a method used to replace variables in an expression with given values to simplify and solve it. This approach is vital when working with expressions like \(b^2 - 4ac\), especially when these values are provided directly. Here’s how it works:

• Identify the variables in your expression (\(b^2 - 4ac\) involves \(b\), \(a\), and \(c\)).
• Directly substitute the given numbers for each variable into the expression. For instance, if \(a = 1\), \(b = 2\), and \(c = 5\), each should replace the respective variable in the expression.
• Perform the arithmetic operations accordingly. Ensure that the order of operations (PEMDAS/BODMAS) is followed to maintain accuracy.

Substitution is especially useful in solving equations implicitly and allows for direct computation of otherwise abstract variables. By practicing substitution, students can become proficient in transforming complex algebraic expressions into solvable arithmetic problems.