Applications and word problems

The richest exam application of this chapter is word problems. A natural English description hides a quadratic, and your job is to extract and solve it.

A three-step plan

1. Define an unknown. Pick a single variable and assign it a meaning with units.
2. Translate the conditions into a quadratic. Use the relations given in the problem.
3. Solve and check. Reject roots that don't make physical sense (negative speeds, negative times, fractional people, etc.).

Common templates

Speed–time–distance. $\text{Time} = \dfrac{\text{Distance}}{\text{Speed}}$. If a vehicle's speed changes by $\Delta v$, the new time changes. Setting the time difference equal to a given quantity produces a quadratic.

Consecutive numbers. Two consecutive integers $n, n + 1$; two consecutive even integers $n, n + 2$. A condition on their product or squares gives a quadratic.

Areas and perimeters. Rectangle of length $l$ and breadth $b$: area $l \cdot b$, perimeter $2(l + b)$. With one relation between $l$ and $b$ (like $l - b = k$ or perimeter $= P$), area becomes a quadratic in one variable.

Age problems. Sometimes ages and their squares appear. The product of two ages, or the difference of squares, often gives a quadratic.

Work problems. Work $=$ rate $\times$ time. A change in rate produces a quadratic in time.

A general strategy: speed–distance

A frequent set-up: a $D$-km journey at speed $v$ takes $D/v$ hours. If speed increases by $\Delta v$, time decreases by $\Delta t = D/v - D/(v + \Delta v)$. Setting this difference $=$ given value yields a quadratic in $v$.

After clearing denominators you typically get a quadratic of the form $a v^2 + b v + c = 0$, and only the positive root is physically meaningful.

Worked examples

Example 1. The sum of the squares of two consecutive odd positive integers is $290$. Find them.

Let the integers be $2n - 1$ and $2n + 1$. Sum of squares $= (2 n - 1)^2 + (2 n + 1)^2 = 8 n^2 + 2 = 290 \Rightarrow n^2 = 36 \Rightarrow n = 6$ (positive).

Integers: $11, 13$.

Example 2. A train travels $480$ km at a uniform speed. If the speed had been $8$ km/h more, it would have taken $3$ hours less for the journey. Find the speed.

Let speed $= v$ km/h. Time $= 480/v$. New time $= 480/(v + 8)$. Difference $= 3$: $\frac{480}{v} - \frac{480}{v + 8} = 3.$ Multiply by $v (v + 8)$: $480 (v + 8) - 480 v = 3 v (v + 8) \Rightarrow 3840 = 3 v^2 + 24 v$. Divide by $3$: $v^2 + 8 v - 1280 = 0$.

Discriminant $= 64 + 5120 = 5184 = 72^2$. $v = (-8 + 72)/2 = 32$ (taking positive root).

Speed $= 32$ km/h.

Example 3. The diagonal of a rectangular field is $60$ m more than the shorter side. The longer side is $30$ m more than the shorter side. Find the sides.

Let shorter side $= x$ m. Then longer $= x + 30$, diagonal $= x + 60$.

By Pythagoras: $x^2 + (x + 30)^2 = (x + 60)^2 \Rightarrow x^2 + x^2 + 60 x + 900 = x^2 + 120 x + 3600 \Rightarrow x^2 - 60 x - 2700 = 0$.

Discriminant $= 3600 + 10800 = 14400$. $\sqrt{D} = 120$. $x = (60 + 120)/2 = 90$ (positive root).

Sides: $90$ m and $120$ m, diagonal $150$ m.

Example 4. Two pipes running together can fill a cistern in $6$ minutes. If one pipe takes $5$ minutes more than the other, find the time each takes to fill the cistern.

Let smaller pipe take $x$ minutes. Larger takes $x + 5$.

Combined rate: $1/x + 1/(x + 5) = 1/6$. So $\dfrac{2 x + 5}{x (x + 5)} = \dfrac{1}{6} \Rightarrow 6(2 x + 5) = x^2 + 5 x \Rightarrow x^2 - 7 x - 30 = 0$.

Factor: $(x - 10)(x + 3) = 0 \Rightarrow x = 10$ (positive). Pipes take $10$ and $15$ minutes.

Example 5. A motorboat whose speed in still water is $18$ km/h takes $1$ hour more to go $24$ km upstream than to return downstream to the same spot. Find the speed of the stream.

Let stream speed $= u$ km/h. Upstream speed $= 18 - u$, downstream $= 18 + u$.

$\dfrac{24}{18 - u} - \dfrac{24}{18 + u} = 1 \Rightarrow \dfrac{24 \cdot 2 u}{(18)^2 - u^2} = 1 \Rightarrow 48 u = 324 - u^2 \Rightarrow u^2 + 48 u - 324 = 0$.

Discriminant $= 2304 + 1296 = 3600$. $u = (-48 + 60)/2 = 6$ (positive). Stream speed $= 6$ km/h.

Try it yourself

1. The product of two consecutive positive integers is $306$. Find the integers.
2. The hypotenuse of a right triangle is $13$ cm. The difference of the other two sides is $7$ cm. Find them.
3. A two-digit number is such that the product of its digits is $18$. When $63$ is subtracted from the number, the digits interchange. Find the number.
4. A train travels at a uniform speed for $360$ km. With speed $5$ km/h more, it takes $1$ hour less. Find the speed.
5. The sum of a number and its reciprocal is $\dfrac{17}{4}$. Find the number.
6. An aeroplane left $30$ minutes later than scheduled; to reach its destination $1500$ km away in time, it increased its speed by $250$ km/h. Find its usual speed.
7. A rectangular park has length $5$ m more than its breadth; its diagonal is $25$ m. Find its dimensions.
8. The sum of the reciprocals of Rehman's ages, $3$ years ago and $5$ years from now, is $1/3$. Find his present age.
9. A pole has to be erected on the boundary of a circular park of $13$ m radius such that the difference of its distances from two diametrically opposite gates is $7$ m. Find the distances.
10. A train travels $63$ km at a uniform speed. The next $72$ km is travelled at a speed $6$ km/h more. The whole journey takes $3$ hours. Find the original speed.

Pitfalls / Insight

• Always check sign / unit / sense of each root. Many quadratic problems give one valid and one rejected root.
• Clear the denominators in time-equation set-ups before applying the formula.
• Re-read the problem before declaring a final answer , many students solve correctly but report the wrong unknown.

Insight. Quadratics make great real-world models because so many quantities trade off as a product (speed and time, length and breadth, two people working together). Spot the product structure, write the relation, get a quadratic, solve.