Problem 11 - Intermediate Algebra

Question

Fill in the blanks. \(b\) is a square root of \(a\) if \(b^{2}= \square\).

Step-by-Step Solution

Answer

The blank should be filled with \(a\).

Step 1: Understand the Concept of Square Root

A number is the square root of another number if squaring it (multiplying it by itself) gives the original number. In this problem, we have to determine what the square root of \(a\) is related to its expression.

Step 2: Interpret the Expression \(b^{2} = \square\)

The expression \(b^{2} = \square\) implies that we need to fill this blank with a function or expression involving \(a\) such that when \(b\) is squared, it equals \(a\).

Step 3: Fill in the Blank

According to the definition of a square root, if \(b\) is a square root of \(a\), then \(b^{2} = a\). This means the expression inside the blank should be \(a\).

Key Concepts

Quadratic Equations

Quadratic equations are an essential part of algebra. They are equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. Solving these equations involves finding the values of \( x \) that make the equation true.
One common method to solve quadratic equations is by using the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Another approach is factoring, which is breaking down the equation into simpler binomials that can be solved using basic arithmetic operations.
Additionally, completing the square is a technique rearranging the equation to form a perfect square trinomial, allowing solutions through the square root method. Understanding these methods helps in solving various real-world problems modeled by quadratic equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They form the backbone of algebra and allow us to represent mathematical situations efficiently.
For example, an expression like \( 3x + 5 \) consists of terms (\( 3x \) and \( 5 \)), where \( 3 \) is the coefficient of \( x \), and \( 5 \) is a constant term.

• Variables, like \( x \) or \( b \), are symbols used to represent unknown values.
• Coefficients are the numbers multiplied by the variables.
• Constants are fixed values that do not change.

Algebraic expressions are manipulated using rules of arithmetic and algebraic identities, like the distributive property, associative property, and factoring. Simplifying these expressions is often crucial in solving equations and inequalities.

Number Properties

Number properties refer to the basic and fundamental characteristics of numbers that help in various mathematical operations and problem-solving. Understanding these properties is crucial for working with different types of numbers, including integers, fractions, and real numbers.
Some key number properties are:

• Commutative Property: For addition and multiplication, the order of the numbers doesn’t change the result. For example, \( a + b = b + a \) and \( ab = ba \).
• Associative Property: The grouping of numbers does not change their sum or product. For example, \( (a + b) + c = a + (b + c) \) and \( (ab)c = a(bc) \).
• Distributive Property: This relates to multiplication over addition or subtraction. For example, \( a(b + c) = ab + ac \).
• Identity Property: The identity for addition is 0, and for multiplication, it is 1, which means adding 0 or multiplying by 1 leaves the original number unchanged.

These properties simplify computations and help in accurately solving equations and understanding more complex algebraic expressions.