There are many situations in science, engineering, economics and other fields where quadratic equations are a vital part of the mathematics used. Some real word problems can be solved using a quadratic equation. We are generally only interested in any real solutions which result. We will look at some of these applications.

We employ the following general problem-solving method when solving worded problems involving quadratic equations.

- Define the terms if required.
- Convert the sentences or sentence parts into algebraic terms.
- Look for keywords such as equal, subtracted from, multiplied by … and so on.
- Write an equation in the form $ax^2+bx+c=0$ form.
- Solve the equation by factorising, completing the square or quadratic formula.
- Decide which solutions are valid
- Answer the question in a sentence.

Any answer we obtain must be checked to see if it is reasonable. For example,

- if we are finding a length it must be positive and we reject any negative solutions.
- if we are finding ‘how many people are present’ then clearly the answer must be an integer.

### Example 1

A rectangle has length $4$ cm longer than its width. If the area is $45$ cm^{2}. Find its width.

If the width is $x$ cm, then the length is $(x+4)$ cm.

\( \begin{align} \displaystyle

x(x+4) &= 45 \\

x^2 + 4x &= 45 \\

x^2 + 4x – 45 &= 0 \\

(x+9)(x-5) &= 0 \\

x+9 &= 0 \text{ or } x-5 = 0 \\

\therefore x &= -9 \text{ or } 5 \\

\end{align} \)

We reject the negative solution as lengths are positive.

Thus the width is $5$ cm.

### Example 2

The product of two consecutive odd numbers is $323$. Find the numbers.

Let two consecutive odd numbers be $x$ and $(x+2)$.

\( \begin{align} \displaystyle

x(x+2) &= 323 \\

x^2 + 2x &= 323 \\

x^2 + 2x – 323 &= 0 \\

(x+19)(x-17) &= 0 \\

x+19 &= 0 \text{ or } x-17 = 0 \\

x &= -19 \text{ or } 17 \\

\therefore x &= 17 \text{ and } x+2 = 19\\

\end{align} \)

We reject the negative solution as odd numbers are positive.

Thus two odd numbers are $17$ and $19$.

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