Problem 11 - Precalculus Student Solutions Manual 5th

Question

Use the following facts. If \(x\) represents an integer, then \(x+1\) represents the next consecutive integer. If \(x\) represents an even integer, then \(x+2\) represents the next consecutive even integer. If \(x\) represents an odd integer, then \(x+2\) represents the next consecutive odd integer. The sum of the squares of two consecutive odd integers is \(202 .\) Find the integers.

Step-by-Step Solution

Answer

The pairs of consecutive odd integers are (9, 11) and (-11, -9).

Step 1: Identify the Variables

Let the first odd integer be represented by the variable x. Since the second integer is the next consecutive odd integer, it is represented as x + 2.

Step 2: Formulate the Equation

According to the problem, the sum of the squares of these two consecutive odd integers is 202. Therefore, the equation can be written as: \[ x^2 + (x + 2)^2 = 202 \]

Step 3: Expand and Simplify the Equation

Expand the square of \( x + 2 \) and simplify the equation: \[ x^2 + (x + 2)^2 = x^2 + x^2 + 4x + 4 = 202 \] Combining like terms, the equation becomes: \[ 2x^2 + 4x + 4 = 202 \]

Step 4: Move All Terms to One Side

To simplify further, move all terms to one side of the equation to set it to zero: \[ 2x^2 + 4x + 4 - 202 = 0 \] This simplifies to: \[ 2x^2 + 4x - 198 = 0 \]

Step 5: Divide the Entire Equation by 2

Divide each term by 2 to make the equation simpler: \[ x^2 + 2x - 99 = 0 \]

Step 6: Solve the Quadratic Equation

To find the roots of the quadratic equation \( x^2 + 2x - 99 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), identify a, b, and c from the equation where a = 1, b = 2, c = -99.

Step 7: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \): \[ 2^2 - 4(1)(-99) = 4 + 396 = 400 \]

Step 8: Find the Roots

Substitute the values back into the quadratic formula to find the two values of x: \[ x = \frac{-2 \pm \sqrt{400}}{2} = \frac{-2 \pm 20}{2} \] So, \( x = 9 \) or \( x = -11 \)

Step 9: Identify the Two Consecutive Odd Integers

If \( x = 9 \), the two integers are 9 and 11. If \( x = -11 \), the two integers are -11 and -9.

Key Concepts

Sum of Squares

The problem involves finding two consecutive odd integers whose squares add up to a specific number, 202. The 'sum of squares' concept here refers to adding the squares of two numbers. Mathematically, the sum of squares of two numbers, say a and b, is represented as: \[ a^2 + b^2 \]. For our problem, if the first odd integer is \( x \), the next consecutive odd integer is \( x + 2 \). So their sum of squares can be written as: \[ x^2 + (x + 2)^2 \]. This equation helps us set up the relationships necessary to solve the problem.

Quadratic Equation

To find the integers, we need to solve a quadratic equation. A quadratic equation is a second-degree polynomial of the form \( ax^2 + bx + c = 0 \). In this problem, we derive the quadratic equation from the sum of squares formula. Starting with: \[ x^2 + (x + 2)^2 = 202 \], expanding and simplifying leads us to: \[ 2x^2 + 4x - 198 = 0 \] and dividing by 2 simplifies it to: \[ x^2 + 2x - 99 = 0 \]. Solving this quadratic equation gives us the values of \( x \) that satisfy our conditions.

Discriminant

The discriminant is a part of the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) used to determine the nature of the roots. It is given by \( b^2 - 4ac \). In our equation \( x^2 + 2x - 99 = 0 \), \( a = 1 \), \( b = 2 \), and \( c = -99 \). Plugging these into the discriminant formula gives: \[ 2^2 - 4(1)(-99) = 4 + 396 = 400 \]. Since the discriminant is a positive perfect square, it indicates two distinct real roots, which we calculate as: \[ \frac{-2 \pm 20}{2} \]. This results in two possible values for \( x \): 9 and -11, leading to the pairs of consecutive odd integers: (9, 11) and (-11, -9).