Problem 344 - Elementary Algebra

Question

The product of two consecutive integers is 42. Find the integers.

Step-by-Step Solution

Answer

The integers are 6, 7 and -7, -6.

Step 1 - Define Variables

Let the two consecutive integers be represented as x and x+1. Since they are consecutive, one integer is one more than the other.

Step 2 - Set Up the Equation

According to the problem, the product of the two integers is 42. So, set up the equation: \[ x(x + 1) = 42 \]

Step 3 - Expand and Rearrange the Equation

Expand the left side of the equation: \[ x^2 + x = 42 \] Then rearrange it to the standard quadratic form: \[ x^2 + x - 42 = 0 \]

Step 4 - Solve the Quadratic Equation

To solve the quadratic equation \( x^2 + x - 42 = 0 \), use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = 1 \) and \( c = -42 \).

Step 5 - Calculate the Discriminant

Compute the discriminant: \[ b^2 - 4ac = 1^2 - 4(1)(-42) = 1 + 168 = 169 \]

Step 6 - Find the Roots

Now, find the roots using the quadratic formula: \[ x = \frac{-1 \pm \sqrt{169}}{2} \] \[ x = \frac{-1 \pm 13}{2} \] This gives two solutions: \[ x = \frac{12}{2} = 6 \] and \[ x = \frac{-14}{2} = -7 \]

Step 7 - Determine Consecutive Integers

Identify the pairs of consecutive integers: If \( x = 6 \), then the integers are 6 and 7. If \( x = -7 \), then the integers are -7 and -6.

Key Concepts

solving quadratic equations

When solving quadratic equations, our goal is to find the values of the variable that make the equation true. Quadratic equations often take the form: \( ax^2 + bx + c = 0 \). For consecutive integers product problems, the quadratic equation is derived from the product equation: \( x(x+1) = 42 \). The steps involve representing the integers with variables, setting up an equation, expanding and rearranging it, and ultimately solving it.

quadratic formula

The quadratic formula is a reliable method to find the roots of a quadratic equation. It is given by: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula can solve any quadratic equation, provided you know the values of \( a \), \( b \), and \( c \). For instance, in the problem \( x(x + 1) = 42 \), after rearranging, we get \( x^2 + x - 42 = 0 \). Here, \( a = 1 \), \( b = 1 \), and \( c = -42 \). Just substitute these values into the quadratic formula to find the solutions.

discriminant calculation

The discriminant helps us determine the nature of the roots of a quadratic equation. It is part of the quadratic formula: \( b^2 - 4ac \). The value of the discriminant tells us:

• If it's positive, we have two distinct real roots.
• If it's zero, we have exactly one real root (a repeated root).
• If it's negative, we have two complex roots.

In our problem, calculating the discriminant involves: \( 1^2 - 4(1)(-42) = 1 + 168 = 169 \). Since 169 is positive, the quadratic equation has two distinct real numbers as roots.

algebraic problem-solving steps

When tackling algebraic problems like this one, follow these structured steps:

• Define Variables: Start by assigning variables to unknown quantities. Here, let the consecutive integers be \( x \) and \( x + 1 \).
• Set Up the Equation: Create an equation based on the problem statement. We have: \( x(x + 1) = 42 \).
• Expand and Rearrange: Make the equation fit the standard quadratic form. Expand to get: \( x^2 + x = 42 \) and then rearrange: \( x^2 + x - 42 = 0 \).
• Solve the Equation: Use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Calculate the discriminant (169) and the roots, \( x = 6 \) and \( x = -7 \).
• Verify: Double-check the results. Here, the pairs are \( 6 \) and \( 7 \) or \( -7 \) and \( -6 \), verifying the original product condition.

Following these steps ensures clarity and accuracy in problem-solving.