Problem 68 - Algebra 1: Concepts and Skills

Question

Evaluate \(\sqrt{b^{2}-4 a c}\) for the given values. $$ a=-5, b=5, c=10 $$

Step-by-Step Solution

Answer

Given the values, the evaluated result of \(\sqrt{b^{2}-4 a c}\) is 15.

Step 1: Substitution

Plug the given values into the formula \(b^{2} - 4ac\), where \(a = -5\), \(b = 5\), and \(c = 10\). Thus, \(5^{2} - 4(-5)(10)\).

Step 2: Simplification

Carry out the arithmetic operation in order to simplify the expression. You get: \(25 + 200 = 225\).

Step 3: Evaluating the square root

Take the square root of 225 to get the final answer.

Key Concepts

Substitution Method

The substitution method is a fundamental concept in algebra, where you replace variables with their actual values. This technique is incredibly useful in solving equations, systems of equations, and evaluating expressions like the one in our exercise.

When you come across an algebraic expression or a formula, such as the square root of a quadratic expression \(\sqrt{b^{2}-4ac}\), the substitution method is your first step. It requires you to take the given values - in this case, \(a=-5\), \(b=5\), and \(c=10\) - and plug them into the appropriate places in the formula.

This means that wherever you see the variable \(a\) in the expression, you replace it with the value \( -5\). Similarly, \(b\) becomes \(5\) and \(c\) becomes \(10\). After substitution, you will end up with a numerical expression free of variables, which can then be simplified using arithmetic operations.

Simplifying Expressions

Simplifying expressions is the process of reducing a complex algebraic expression into its simplest form. This makes it easier to interpret or to further operate on the expression. In our example, after substituting the values into the expression \(5^{2} - 4(-5)(10)\), the next logical step is simplification.

This involves performing arithmetic operations such as addition, subtraction, multiplication, and division to eliminate parentheses and combine like terms. In the expression above, we first square \(b\), which is 5, resulting in \(25\). Then, we multiply \(4\), \(a\), and \(c\) together, keeping in mind that multiplying two negatives gives a positive result, giving us \(200\).

We then add the results of these operations \(25 + 200\) to arrive at \(225\), a much simpler expression. This simplified form can be used to carry out further operations, such as finding the square root as required in the problem.

Arithmetic Operations

Arithmetic operations include the basic mathematical processes we use to carry out calculations. These operations include addition, subtraction, multiplication, division, and extracting roots, which are essential for simplifying expressions and solving problems.

In the context of our problem, after simplifying the substituted expression to \(225\), we have essentially performed the arithmetic operations of squaring a number and multiplying, followed by addition.

The last step, evaluating the square root of \(225\), is another arithmetic operation. To evaluate \(\sqrt{225}\), you need to determine the number, which, when multiplied by itself, gives \(225\). This number is \(15\), as \(15 \times 15 = 225\).

The ability to carry out arithmetic operations swiftly and accurately is crucial in algebra, not only to simplify expressions but also to solve equations and understand other mathematical concepts.