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Example

Finding Two Consecutive Odd Integers Whose Product Is 195

Apply the seven-step problem-solving strategy to find two consecutive odd integers when their product is known, using the Quadratic Formula to solve the resulting equation.

Problem: The product of two consecutive odd integers is 195. Find the integers.

  1. Read the problem.
  2. Identify what to find: two consecutive odd integers.
  3. Name the unknowns: Let nn = the first odd integer. Then n+2n + 2 = the next consecutive odd integer.
  4. Translate into an equation: "The product of the first odd integer and the second odd integer is 195" becomes:

n(n+2)=195n(n + 2) = 195

  1. Solve the equation. Distribute on the left side:

n2+2n=195n^2 + 2n = 195

Subtract 195 from both sides to obtain standard form:

n2+2n−195=0n^2 + 2n - 195 = 0

Identify the coefficients: a=1a = 1, b=2b = 2, c=−195c = -195. Substitute into the Quadratic Formula:

n=−2±22−4(1)(−195)2(1)=−2±4+7802=−2±7842n = \frac{-2 \pm \sqrt{2^2 - 4(1)(-195)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 780}}{2} = \frac{-2 \pm \sqrt{784}}{2}

Since 784=28\sqrt{784} = 28:

n=−2±282n = \frac{-2 \pm 28}{2}

Split into two solutions:

n=−2+282=262=13orn=−2−282=−302=−15n = \frac{-2 + 28}{2} = \frac{26}{2} = 13 \quad \text{or} \quad n = \frac{-2 - 28}{2} = \frac{-30}{2} = -15

Because there are two values of nn, two pairs of consecutive odd integers satisfy the condition:

  • If n=13n = 13, then n+2=15n + 2 = 15, giving the pair 13 and 15.
  • If n=−15n = -15, then n+2=−13n + 2 = -13, giving the pair −15-15 and −13-13.
  1. Check: Are 13 and 15 consecutive odd integers? Yes. Is 13⋅15=19513 \cdot 15 = 195? Yes ✓. Are −13-13 and −15-15 consecutive odd integers? Yes. Is (−13)(−15)=195(-13)(-15) = 195? Yes ✓.
  2. Answer: The two consecutive odd integers whose product is 195 are 13, 15 and −13-13, −15-15.

This example extends the consecutive-integer product technique to odd integers and uses the Quadratic Formula instead of factoring. Unlike the analogous problem for general consecutive integers (which was solved by factoring), this equation n2+2n−195=0n^2 + 2n - 195 = 0 is solved with the Quadratic Formula because the constant −195-195 is harder to factor by inspection. Both pairs of solutions are valid because multiplying two negative odd integers also yields a positive product.

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Updated 2026-04-21

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References

Tags

OpenStax

Elementary Algebra @ OpenStax

Ch.10 Quadratic Equations - Elementary Algebra @ OpenStax

Algebra

Math

Prealgebra

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