When a Word Problem Gives Two Answers, How Do You Know Which One Is Real?

Every problem in this chapter follows the same four steps:

- Let the unknown quantity be $x$, and write the other quantities in terms of $x$
- Form an equation from the condition in the problem
- Solve the quadratic equation
- Check both roots against the situation and reject any that are impossible, giving the reason

A length, speed or age cannot be negative, so one root is often rejected.

**Write consecutive integers as $x$ and $x+1$ and a reciprocal as $\frac{1}{x}$, then form and solve the equation.

Worked example 1.** The product of two consecutive positive integers is $132$.

$$x(x+1)=132x^2+x-132=0(x+12)(x-11)=0$$

**$x=-12$ is rejected because the integers are positive.** The integers are $11$ and $12$.

Worked example 2. A number plus its reciprocal is $\frac{10}{3}$.

$$x+\frac{1}{x}=\frac{10}{3}3x^2-10x+3=0(3x-1)(x-3)=0x=3\text{ or }\frac{1}{3}$$

Both roots are valid here.

An everyday example. **A book lies open, and the product of the two facing page numbers is $420$.** Then $x(x+1)=420$ gives $x=20$: pages $20$ and $21$.

The substance. Reject a root only when the problem rules it out — sometimes both answers are correct.

**Write future ages as $x+n$, and a two-digit number with digits $x$ and $y$ as $10x+y$.

Worked example 1 — ages.** A mother is $26$ years older than her daughter. In $3$ years, the product of their ages will be $360$.

$$(x+3)(x+29)=360x^2+32x-273=0$$
$$x=\frac{-32+\sqrt{1024+1092}}{2}=\frac{-32+46}{2}=7$$

The negative root is rejected. The daughter is $7$ and the mother $33$; check $10\times 36=360$.

Worked example 2 — two-digit number. A two-digit number is $4$ times the sum of its digits and $2$ times the product of its digits.

$$10x+y=4(x+y)\Rightarrow y=2x10x+y=2xy\Rightarrow 12x=4x^2$$

**$x=0$ is rejected, since a tens digit cannot be zero**, so $x=3$, $y=6$ and the number is $36$.

An everyday example. Family age puzzles at a birthday party often hide exactly this kind of quadratic.

The substance. Always check the final answer in the original words, not just in the equation.

**Use time $=$ distance $÷$ speed to write each time in terms of $x$, set up the stated difference or total, and remember that upstream speed is the boat's speed minus the stream's speed while downstream speed is their sum.

Worked example 1 — train.** A train covers $480$ km. If its speed were $8$ km/h more, it would take $3$ hours less.

$$\frac{480}{x}-\frac{480}{x+8}=33840=3x(x+8)x^2+8x-1280=0(x+40)(x-32)=0$$

**Speed $=32$ km/h** (a speed cannot be negative). Check: $\frac{480}{32}=15$ h and $\frac{480}{40}=12$ h.

Worked example 2 — stream. A boat's speed in still water is $15$ km/h. It goes $30$ km upstream and returns in $4.5$ hours. Find the speed of the stream $x$.

$$\frac{30}{15-x}+\frac{30}{15+x}=4.5900=4.5(225-x^2)x^2=25$$

**Stream speed $=5$ km/h.** Check: $\frac{30}{10}+\frac{30}{20}=3+1.5=4.5$ h.

An everyday example. A river ferry takes longer against the current than with it.

The boundary case. **The stream speed must be less than $15$ km/h**, otherwise the boat could never move upstream.

Form the equation from area formulas or work done per hour, solve, and reject impossible roots with a reason.

Worked example 1 — field. A rectangular field has perimeter $70$ m and area $300{\text{ m}}^2$.

$$x(35-x)=300x^2-35x+300=0(x-15)(x-20)=0$$

**The sides are $20$ m and $15$ m.

Worked example 2 — path.** A park $60$ m by $40$ m has a path of uniform width inside along its edges. The remaining lawn has area $1500{\text{ m}}^2$.

$$(60-2x)(40-2x)=15004x^2-200x+900=0x^2-50x+225=0(x-5)(x-45)=0$$

**$x=45$ is rejected, because $40-90$ would make the lawn's width negative.** The path is $5$ m wide.

Worked example 3 — pipes. One pipe fills a tank in $5$ hours more than another. Together they fill it in $6$ hours.

$$\frac{1}{x}+\frac{1}{x+5}=\frac{1}{6}12x+30=x^2+5x{x}^2-7x-30=0(x-10)(x+3)=0$$

**The pipes take $10$ and $15$ hours**; $x=-3$ is rejected as time cannot be negative.

An everyday example. Two taps filling an overhead water tank on a roof follow the same work equation.

The substance. The reason for rejection carries marks — write it, do not just cross the root out.