Forming quadratic equations from words

Many problems about numbers, ages, speeds and amounts lead to a quadratic equation. Choose a letter for one unknown, write the other quantities in terms of it, and turn the sentence into an equation.

Why: a product of two unknowns, or the square of an unknown, gives an \(x^2\)-term. The equation may have two roots, but the problem can rule one out, for example when a number must be positive or a whole number.

Method

1. Let \(x\) stand for one quantity and state what it is, with its unit.
2. Write the other quantities in terms of \(x\).
3. Form the equation from the information given and rearrange it to the form \(ax^2+bx+c=0\).
4. Solve, reject any root that does not fit the situation and give a reason, then answer the question in words.

Example

The product of two consecutive positive even numbers is \(168\). Find the two numbers.

Let the smaller number be \(x\). The next even number is \(x+2\).

\(\begin{aligned} x(x+2) &= 168 \\ x^2+2x-168 &= 0 \\ (x+14)(x-12) &= 0 \\ x=-14 \quad &\text{or}\quad x=12 \end{aligned}\)

Reject \(x=-14\) because the numbers are positive. The numbers are \(12\) and \(14\).

Check: \(12\times14=168\) and both numbers are even. A common error is to stop at the roots; the question asks for both numbers, not just \(x\).