NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Quadratic graphs and equations

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Two friends, Alex and Jordan, are comparing walking routes from their home in Melbourne to a park. Alex takes a path that follows a quadratic distance-time relationship given by d = t^2 - 22t + 59, where d is the distance in metres from home and t is the time in seconds.

Jordan takes a different route that follows d = t^2 - 22t + 64. At what time t (in seconds) do both routes have the same distance from home?

Give your answer as an exact value.

Concept illustrations

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The graph y = x² − 9 crosses the x-axis at −3 and 3. These are the solutions of x² − 9 = 0; the vertex is (0,−9).
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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The problem gives two quadratic distance-time equations: d_Alex = t^2 - 22t + 59 and d_Jordan = t^2 - 22t + 64. We are to find the time t when both distances are equal.
  2. Set the two expressions for d equal to each other: t^2 - 22t + 59 = t^2 - 22t + 64.
  3. Subtract t^2 from both sides: -22t + 59 = -22t + 64.
  4. Add 22t to both sides: 59 = 64.
  5. This results in a contradiction: 59 = 64 is false. Therefore, there is no value of t for which the two expressions are equal.
  6. Since the two quadratic functions have identical t^2 and t coefficients but different constant terms, they are parallel parabolas shifted vertically. They never intersect.
  7. The absence of a solution means that the two routes never have the same distance from home at any time t.
  8. Therefore, there is no real value of t that satisfies the condition.
  9. The question asks for the time t when both routes have the same distance from home. Since no such t exists, the answer is that no solution exists.
  10. However, the question specifies to give an exact value. Since no value satisfies the equation, the correct response is that no such time exists.
  11. Rechecking: the two equations differ only in the constant term (59 vs 64). The difference is constant: (t^2 - 22t + 64) - (t^2 - 22t + 59) = 5. So d_Jordan is always 5 m greater than d_Alex. They never equal.
  12. Thus, there is no time t at which both routes have the same distance from home. The answer is that no such time exists.

Reference answer: No solution exists

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