Solving problems with quadratics

Writing quadratic equations from area, number and motion problems, solving them and rejecting solutions that do not make sense.

Method

Example 1●●

A rectangle's length is $3$ m more than its width, and its area is $40$ m². Find the width.

$w(w + 3) = 40$, so $w^2 + 3w - 40 = 0$.

$(w + 8)(w - 5) = 0$, so $w = -8$ or $w = 5$.

A width cannot be negative, so the width is $5$ m.

Example 2●●

A ball's height is $h = 20t - 5t^2$ metres after $t$ seconds. When does it land?

$20t - 5t^2 = 0$, so $5t(4 - t) = 0$.

$t = 0$ (thrown) or $t = 4$. It lands after $4$ s.

Height of the ball

Drag along the curve. The ball is highest at t = 2 s, halfway between the two zeros.

Flashcards

Negative length?

Reject it.

Lands when

$h = 0$

Consecutive numbers

$n$ and $n + 1$

Practice

Length $3$ m more than width, area $40$ m². Width?
$h = 20t - 5t^2$. When does the ball land (s)?
Using $h = 20t - 5t^2$, find the maximum height (at $t = 2$).
Two consecutive positive whole numbers multiply to $56$. The smaller?
A square's area is $36$ cm². Its side?
A number squared minus twice the number is $48$. Find the positive number.
For $h = 20t - 5t^2$, when is the ball at height $15$ m? Enter both times.
A triangle has base $x$ cm and height $(x - 6)$ cm, and area $20$ cm². Find $x$.