Problems with quadratic equations

Setting up and solving quadratic equations from area, motion, number and finance problems, and interpreting which solutions make sense.

Examples

Example 1●●

The product of two consecutive positive numbers is $30$. Find them.

$n(n + 1) = 30$, so $n^2 + n - 30 = 0$.

$(n + 6)(n - 5) = 0$; $n = 5$ (positive). The numbers are $5$ and $6$.

Example 2●●●

A garden is $3$ m longer than it is wide, with area $50$ m². Find its width to 2 d.p.

$w(w + 3) = 50$, so $w^2 + 3w - 50 = 0$.

$w = \dfrac{-3 + \sqrt{9 + 200}}{2} \approx 5.73$ m (reject the negative).

Flashcards

Reject solutions that are

Negative lengths, times or counts

Doesn't factorise?

Use the quadratic formula.

Area of rectangle

$l \times w$

Practice

Two consecutive positive numbers multiply to $30$. The smaller?
Garden $3$ m longer than wide, area $50$ m². Width (2 d.p.)?
$h = 30t - 5t^2$. When is the ball $45$ m high?
Using $h = 30t - 5t^2$, when does the ball land?
A square's side is increased by $2$ cm, making the area $100$ cm². Original side?
A number plus its square is $156$. Find the positive number.
A right triangle has legs $x$ and $x + 2$, and hypotenuse $\sqrt{52}$. Find $x$.
A rectangle has perimeter $34$ and area $60$. Find its length (the longer side).