Problem 62 - Algebra 1: Concepts and Skills

Question

Evaluate the radical expression when $a=-1 \text { and } b=5. $$\sqrt{b^{2}-11 a}$$

Step-by-Step Solution

Answer

The value of the expression \( \sqrt{b^{2}-11 a} \) when \( a=-1 \) and \( b=5 \) is 6.

Step 1: Substitution

Substitute \( a = -1 \) and \( b = 5 \) into the expression \( \sqrt{b^{2} - 11a} \). This gives us\[ \sqrt{5^{2} - 11(-1)} \].

Step 2: Simplify Inside the Radical

First, simplify the expression inside the square root. This gives us \( \sqrt{25 - (-11)} \).

Step 3: Continue Simplifying

Continue simplifying under the radical. We are subtracting a negative, which is equivalent to adding a positive, so the expression becomes \( \sqrt{25 + 11} \).

Step 4: Final Simplification

Simplify the sum under the square root to obtain \[ \sqrt{36} \].

Step 5: Evaluate the Square Root

Evaluate \( \sqrt{36} \) to get 6.

Key Concepts

Square Root

The square root is a fundamental concept in mathematics that involves finding a number which, when multiplied by itself, gives the original number. We denote the square root of a number with the symbol \( \sqrt{} \). For example, the square root of 36 is expressed as \(\sqrt{36} = 6\) because \(6 \times 6 = 36\).

• Square roots are typically applied to non-negative numbers.
• The number under the square root sign is known as the "radicand."
• Square roots are essential in solving quadratic equations and occur frequently in geometry and algebra.

When dealing with square roots in algebraic expressions, it's important to simplify them as much as possible, especially when they are part of more complex calculations.

Simplification

Simplification is crucial when working with algebraic expressions, particularly those involving radicals like square roots. It involves reducing expressions to their simplest form.

• Simplification makes expressions easier to understand and solve.
• It often involves basic operations like addition, subtraction, multiplication, and division.

In the context of our exercise, simplification involves both the expression inside the square root and the final calculation of the square root itself. For instance, converting \(\sqrt{25 - (-11)}\) to \(\sqrt{25 + 11}\) is a simplification step where we recognize that subtracting a negative is equivalent to adding the corresponding positive number.

Understanding how to simplify expressions under radicals helps prevent errors and leads to more straightforward solutions.

Substitution

Substitution is a fundamental technique in algebra used to evaluate expressions by replacing variables with their known values. When given specific values for variables, substitution allows us to turn an abstract expression into a concrete number.

• This is essential for solving equations or evaluating expressions.
• In our problem, we replace \(a\) with \(-1\) and \(b\) with \(5\).

By substituting \(b = 5\) and \(a = -1\) into \(\sqrt{b^2 - 11a}\), the expression becomes \(\sqrt{5^2 - 11(-1)}\). This transforms the equation to something we can readily solve. Understanding substitution can significantly streamline problem-solving processes.

Negative Numbers

Negative numbers play a critical role in mathematics, representing values less than zero. They are denoted with a minus sign (\(-\)). When incorporated into expressions, especially under square roots, special attention is needed to avoid errors.

• In some mathematical contexts, such as arithmetic operations, they behave differently than positive numbers.
• Subtraction of a negative number is equivalent to addition. For instance, \(25 - (-11)\) becomes \(25 + 11\).

In algebraic expressions, wrong handling of negative numbers can lead to calculation mistakes. Understanding their properties ensures accurate simplification of expressions and evaluations, such as in this exercise where subtracting \(-11\) effectively becomes adding 11.