Problem 60 - Introductory Algebra for College Students

Question

The square of the difference between a number and 7 is 16 Find the number \((s)\)

Step-by-Step Solution

Answer

The numbers that satisfy the given problem are 11 and 3

Step 1: Write the algebraic statement of the problem

First, write down the algebraic expression that represents the problem: \((s-7)^2 = 16\)

Step 2: Apply square roots to both sides of the equation

In order to remove the square from '(s-7)^2', apply square root to both sides. It results in the equation: \(s-7 = \sqrt{16}\) and \(s-7 = -\sqrt{16}\). The reason for the 'plus and minus' on the right side is that when we square a number, we lose the information about its sign, since squaring both positive and negative numbers results in a positive number.

Step 3: Compute the square root

Square root of 16 is 4. So, the two equations are: \(s-7 = 4\) and \(s-7 = -4\)

Step 4: Solve for 's'

Solving for 's' in both equations, data from step 3 yields \( s = 7+4=11\) and \(s = 7-4=3\)

Key Concepts

Quadratic Equations

Quadratic equations are a special type of polynomial equations, and they have the general form of:
\[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants, and the highest exponent of the variable \( x \) is 2.

In the context of algebraic equations, the task is often to find the value of \( x \) that satisfies these equations.
Quadratic equations differ from linear equations because they can have two solutions, and these solutions might be real or complex numbers.

• Characteristics: The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \( a \).
• Real-life Applications: They are used in physics for calculating trajectories, in business for profit predictions, and in many other fields of science and engineering.

Square Roots

The square root is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. The square root is represented using the square root symbol \(\sqrt{}\).
For example, \(\sqrt{16} = 4\), because \(4 \times 4 = 16\).

Moreover, taking the square root of both sides of an equation helps in solving quadratic equations, especially when one side is a perfect square.

• Sign Importance: When solving equations, it is crucial to consider both the positive and negative roots. This is because both positive and negative numbers give the same result when squared, e.g., \((-4)^2 = 16\) and \(4^2 = 16\).
• Application in Algebra: Using square roots to "undo" squares simplifies finding solutions to quadratic equations and other mathematical expressions.
• Limitations: Square roots are only defined for non-negative numbers in the set of real numbers, which is important in ensuring that solutions are real and applicable.

Integers

Integers are a fundamental number set in mathematics. They include all positive whole numbers, negative whole numbers, and zero. In notation form, integers are represented by \( \mathbb{Z} \).

Integers play a crucial role in algebra and many other areas of math due to their simplicity and applicability in modeling real-world quantities.

• Types of Numbers: Integers are part of the broader group of numbers that includes rational numbers, irrational numbers, and real numbers.
• Usefulness: In algebra, the straightforward properties of integers, such as their closure under addition, subtraction, and multiplication, make them an ideal number system for solving equations.
• Basic Operations: When working with integers in equations, note that operations such as addition, subtraction, and multiplication will yield integers, while division might not.

Mathematical Problem-Solving

Mathematical problem-solving is the process of finding solutions to complex mathematical questions or equations. It involves a strategic understanding of mathematical concepts, logical thinking, and the application of appropriate techniques.

In the context of algebraic equations like quadratic ones, it might involve several steps such as translating words into equations, isolating variables, and simplifying expressions.

• Strategy Use: Problem-solving often starts by understanding what the question is asking, identifying known and unknown quantities, and forming equations accordingly.
• Solution Methods: For quadratic equations, strategies such as factoring, using the quadratic formula, and completing the square are essential tools.
• Verification: After finding a solution, it's crucial to check the correctness by substituting back into the original problem to ensure consistency and completeness.
• Practice: Regular practice with different problem types enhances skills and confidence in using different methods effectively.