Problem 67 - Algebra 1: Concepts and Skills

Question

Evaluate \(\sqrt{b^{2}-4 a c}\) for the given values. $$ a=-2, b=8, c=-8 $$

Step-by-Step Solution

Answer

The evaluated result of the expression for the given values is 0.

Step 1: Substitute values of a, b, c

The first step is to substitute the given values into the expression: \(\sqrt{8^{2}-4(-2)(-8)}\).

Step 2: Perform Multiplication and Addition

Perform the multiplication inside the square root expression first: \(\sqrt{64-64}\).

Step 3: Evaluate square root

Upon performing the subtraction inside the square root, we get \(\sqrt{0}\). The square root of 0 is 0.

Key Concepts

Substitution in Algebraic Expressions

The procedure of substitution is fundamental in algebra, and it is a pivotal step when solving various mathematical problems. It involves replacing variables with their numerical values to simplify or evaluate expressions. In the example, to evaluate \(\sqrt{b^{2}-4ac}\), we substitute \(a=-2\), \(b=8\), and \(c=-8\) to the expression. \(
\)First, understand what each symbol represents, then carefully replace each variable with the given number. Ensuring that signs are correct, multiplication is done before addition and subtraction (following the order of operations), and considering any exponents, are all part of accurate substitution. This precise and methodical approach allows us to transform the abstract expression into a concrete numerical value that can be further evaluated.

Quadratic Formula

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\), is a powerful tool for finding the roots of quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). While the exercise at hand did not require finding the roots of a quadratic equation, the expression \(\sqrt{b^{2}-4ac}\) is indeed part of the quadratic formula, specifically under the square root is known as the discriminant. \(
\)Understanding the discriminant is crucial as it determines the nature of the roots of a quadratic equation—real and distinct, real and equal, or complex. When you come across the expression \(\sqrt{b^{2}-4ac}\), it is often in the context of solving quadratic equations, and its evaluation is a significant step in that process.

Simplifying Radicals

Simplifying radicals involves reducing the expression inside the square root to its simplest form. This process often requires identifying and extracting perfect squares, cubes, etc., from under the radical. Simplifying can make equations more manageable and easier to understand. \(
\)In the given exercise, after performing the substitution and arithmetic operations, we obtain \(\sqrt{0}\). The square root of zero is a unique case because zero is the only number that equals its square root—essentially, it's already in its simplest form. Generally, if the square root of a positive integer is not an integer, the challenge is to break down the number into its prime factors and look for pairs to bring outside of the radical. It's essential to recognize when a radical cannot be simplified further to prevent unnecessary computations and errors.