Problem 63 - Algebra A Combined Function

Question

The product of a number and 4 less than the number is \(96 .\) Find the number.

Step-by-Step Solution

Answer

The number is 12 or -8.

Step 1: Set up the equation

Let the number be \(x\). Then \(x(x-4) = 96\).
\(x^2 - 4x - 96 = 0\)

Step 2: Factor or use the quadratic formula

We need two numbers that multiply to \(-96\) and add to \(-4\): \(-12\) and \(8\).
\((x-12)(x+8) = 0\)

Step 3: Solve

\(x = 12\) or \(x = -8\).
Check: \(12 \times 8 = 96 \checkmark\) and \((-8)(-12) = 96 \checkmark\).
The number is 12 or \(-8\).

Key Concepts

solving quadratics

When we talk about solving quadratics, we are dealing primarily with polynomial equations that include terms up to the second degree. In simpler terms, a quadratic equation typically looks like this: \( ax^2 + bx + c = 0 \). However, not all quadratic problems start or end similarly. In our exercise, we're given a problem that results in a quadratic equation upon set-up. You begin by setting the equation based on the context:"the product of a number (let's call it \( x \)) and 4 less than the number is 96". This setup translates to an algebraic equation: \( x(x-4) = 96 \). To solve this quadratic equation, you need to expand and reorganize it in the standard quadratic form: \( x^2 - 4x = 96 \) becomes \( x^2 - 4x - 96 = 0 \). After which, you can use various methods to find the solution such as:

• Factoring, where you find two numbers that multiply to \( c \) (in -96) and add to \( b \) (-4).
• Quadratic formula, which is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This reliable method gives the roots directly.
• Completing the square, where you manipulate the equation into a perfect square trinomial.

Each method has its use, and choosing one depends on what's easiest given the specific equation.

variable representation

Variable representation is the foundation for solving algebraic expressions and equations. In simple terms, a variable is a symbol that stands in for an unknown value that you're trying to find.In our problem, the unknown number is represented by the variable \( x \). When you are modeling the situation described in the problem, using a variable makes it easy to relate different values together in one cohesive equation. We started by letting \( x \) be the number. Then, to represent "4 less than the number," we express that quantity as \( x - 4 \).This form of representation helps in setting up equations because it makes manipulating and solving them straightforward as you perform operations to isolate and find the value of the variable. Using simple variable representations keeps the solving process clear, particularly in problems that lead to quadratics.

algebraic expressions

Algebraic expressions consist of numbers, variables, and operations. They form the backbone of algebra, enabling us to represent real-world situations with clarity and precision. An understanding of how to manipulate these is crucial when working on math problems.In our original exercise, we formed an algebraic expression to describe the problem scenario: \( x(x-4) \). This expression describes the product of a number, \( x \), and four less than that number, \( x-4 \).Here are some key uses of algebraic expressions in solving our problem:

• Translate words into mathematical terms, bridging the gap between storytelling and numerical computation.
• Facilitate forming equations that can be solved using various algebraic techniques.
• Be manipulated algebraically to simplify, factor, or extend relationships to solve quadratic equations completely.

Understanding how to set up these expressions correctly from a narrative or description is vital because it lays the groundwork for further steps in solving equations, finding variable values, and completing mathematical proofs.