Problem 18 - Algebra 2 and Trigonometry

Question

Brad is 3 years older than Francis. The product of their ages is 154. Determine their ages.

Step-by-Step Solution

Answer

Francis is 11 years old and Brad is 14 years old.

Step 1: Define Variables

Let's denote Francis's age as \(x\). Since Brad is 3 years older than Francis, Brad’s age can be expressed as \(x + 3\).

Step 2: Set Up the Equation

According to the problem, the product of their ages is 154. Therefore, we can write the equation: \(x(x + 3) = 154\).

Step 3: Expand and Form a Quadratic Equation

Expand the equation to get a quadratic equation: \(x^2 + 3x = 154\).

Step 4: Rearrange the Quadratic Equation

Rearrange the quadratic equation to set it to zero: \(x^2 + 3x - 154 = 0\).

Step 5: Solve the Quadratic Equation

To solve the quadratic equation \(x^2 + 3x - 154 = 0\), we need to find two numbers whose product is -154 and sum is 3. These numbers can be found by factoring, which results in \((x - 11)(x + 14) = 0\). Therefore, \(x = 11\) or \(x = -14\).

Step 6: Determine Feasible Solution

Since Francis's age cannot be negative, we discard \(x = -14\). Hence, Francis is 11 years old.

Step 7: Calculate Brad's Age

Since Brad is 3 years older than Francis, Brad’s age is \(11 + 3 = 14\).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, which means they include terms up to the square of the unknown variable. A standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the unknown variable.
What makes quadratic equations unique is their parabolic graph shape. Solving these equations involves finding the values of \(x\) that satisfy the equation.
These solutions can be found using various methods, such as:

• Factoring
• Quadratic formula \(\left( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \right)\)
• Completing the square

In the context of our age problem, the quadratic equation \(x^2 + 3x - 154 = 0\) is derived from the condition that the product of the ages is 154. From this equation, we determine potential values for Francis's age.

Factoring

Factoring is one of the methods used to solve quadratic equations. It involves expressing the quadratic expression as a product of its linear factors. When a quadratic equation is factorable, it means we can rewrite it as

• \((px + q)(rx + s) = 0\)

where \(px + q\) and \(rx + s\) are linear expressions. Finding these factors is key to solving the equation, as by the zero-product property, if the product of two expressions is zero, at least one of the expressions must be zero.
In our age problem, the equation \(x^2 + 3x - 154 = 0\) is factorable, leading to the factors \((x - 11)(x + 14) = 0\). Solving these gives us the potential ages for Francis: \(x = 11\) or \(x = -14\). Since age can't be negative, \(x = 11\) is the solution, indicating Francis is 11 years old.

Age Problems in Algebra

Age problems in algebra involve using mathematical equations to determine the ages of individuals based on given conditions. These problems often rely on relationships between ages, such as differences, sums, or products, and involve setting up equations to represent these relationships.
In our problem, we know that Brad is 3 years older than Francis and the product of their ages equals 154.
The main steps include:

• Defining variables for the unknown ages, e.g., let \(x\) be Francis's age.
• Using these variables to express relationships, e.g., Brad's age is \(x + 3\).
• Setting up and solving equations based on the problem's conditions.

For Brad and Francis, the equation \(x(x + 3) = 154\) helped us establish a quadratic equation whose solution revealed their ages, reinforcing the importance of translating words into mathematical expressions in age problems.