Problem 76 - Algebra 1

Question

Two numbers have a geometric mean of \(12 .\) One number is 32 more than the other. Find the numbers.

Step-by-Step Solution

Answer

The two numbers are 4 and 36.

Step 1: Set Up the Equations

First, establish two equations based on the given conditions: \[x * y = (12^2) = 144\] and \[y = x + 32\].

Step 2: Substitute Second Equation into First

Replace \(y\) in the first equation with \(x+32\) from the second equation to get a single equation in terms of \(x\): \[x * (x + 32) = 144\].

Step 3: Simplify the Equation

Expand and simplify this equation to obtain a quadratic equation: \(x^2 + 32x - 144 = 0\).

Step 4: Solve the Quadratic Equation

This quadratic can be factored into \((x-4)*(x+36)=0\). This gives two possible solutions for \(x\), which are \(x=4\) and \(x=-36\). However, since we are looking for a positive number, the solution \(x=-36\) can be discarded.

Step 5: Find the Second Number

Substitute \(x=4\) into the equation \(y=x+32\) to find the value of \(y\). You get \(y=36\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations form an integral part of algebra and are recognizable by their standard form, which is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(a \eq 0\). They are called quadratic because \(quad\) means 'square' in Latin, and the highest degree of the variable \(x\) is 2.

In our exercise, we deal with finding two numbers whose geometric mean is 12. The relationship between these two numbers can be expressed as a quadratic equation, \(x^2 + 32x - 144 = 0\). This equation was derived by substituting the condition that one number is 32 more than the other and that their product equals the square of the geometric mean, which is \(144\).

The methods to solve a quadratic equation include factoring, using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), and completing the square method. In our example, the equation is factored into \(x-4)(x+36) = 0\) to find the values of \(x\). Factoring is often the simplest method when it is possible, as it was in this case.

Algebraic Problem Solving Strategies

Algebraic problem solving involves a set of strategic steps to simplify and solve equations. The key is to identify the relationships between the quantities involved and express them in algebraic form.

The process typically starts with defining variables to represent unknown quantities. In our geometric mean problem, variables \(x\) and \(y\) were used to represent the two unknown numbers. We then formulate equations based on the given conditions. Substitution is a common strategy where an expression for one variable is replaced into another equation, reducing the number of variables and simplifying the problem.

Breaking Down Complex Problems

Complex problems are often made up of simpler parts. By breaking down a complex problem into simpler components, we're able to tackle each part systematically. For the geometric mean question, we first established the relationship between the two numbers and the geometric mean, and then addressed the condition that one number is greater than the other by a specific amount.

Strategies like these turn seemingly difficult algebraic problems into manageable equations that can be solved through familiar methods such as factoring, graphical solutions, or computational tools.

Factoring Quadratic Expressions

Factoring is the process of breaking down a complicated expression into a product of simpler ones. In the context of quadratic expressions, it involves finding two binomials that multiply together to give the original quadratic equation.

Take the equation \(x^2 + 32x - 144 = 0\) from our textbook example. Factoring involves looking for two numbers that multiply to \(ac\) (which is -144) and add to \(b\) (which is 32). We found that 4 and 36 satisfy these conditions, since \(4 \times 36 = 144\) and \(4 + 36 = 40\). The equation then factors into \(x - 4)(x + 36) = 0\).

Significance of Factoring

Factoring is valuable as it transforms the problem into a simpler form where the Zero Product Property can be applied, stating that if a product is zero, then at least one of the factors must be zero. This property is crucial because it provides a clear path to finding the solution for \(x\), which, in our problem, led to finding the two numbers with a geometric mean of 12.