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A Quadratic Equation looks like this:

Quadratic equations pop up in many real world situations!

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Here we have collected some examples for you, and solve each using different methods:

Each example follows three general stages:

Take the real world description and make some equations
Solve!
Use your common sense to interpret the results

Balls, Arrows, Missiles and Stones

When you throw a ball (or shoot an arrow, fire a missile or throw a stone) it goes up into the air, slowing as it travels, then comes down again faster and faster ..

.. and a Quadratic Equation tells you its position at all times!

Example: Throwing a Ball

A ball is thrown straight up, from 3 m above the ground, with a velocity of 14 m/s. When does it hit the ground?

Ignoring air resistance, we can work out its height by adding up these three things:
(Note: t is time in seconds)

The height starts at 3 m:	3
It travels upwards at 14 meters per second (14 m/s):	14t
Gravity pulls it down, changing its position by about 5 m per second squared:	−5t²
(Note for the enthusiastic: the -5t² is simplified from -(½)at² with a=9.8 m/s²)

Add them up and the height h at any time t is:

h = 3 + 14t − 5t²

And the ball will hit the ground when the height is zero:

3 + 14t − 5t² = 0

Which is a Quadratic Equation !

In 'Standard Form' it looks like:

−5t² + 14t + 3 = 0

It looks even better when we multiply all terms by −1:

5t² − 14t − 3 = 0

Let us solve it ..

There are many ways to solve it, here we will factor it using the 'Find two numbers that multiply to give a×c, and add to give b' method in Factoring Quadratics:

a×c = −15, and b = −14.

The factors of −15 are: −15, −5, −3, −1, 1, 3, 5, 15

By trying a few combinations we find that −15 and 1 work (−15×1 = −15, and −15+1 = −14)

Rewrite middle with −15 and 1:5t²− 15t + t − 3 = 0

Factor first two and last two:5t(t − 3) + 1(t − 3) = 0

And the two solutions are:5t + 1 = 0 or t − 3 = 0

The 't = −0.2' is a negative time, impossible in our case.

The 't = 3' is the answer we want:

The ball hits the ground after 3 seconds!

Here is the graph of the Parabola h = −5t² + 14t + 3

It shows you the height of the ball vs time

Some interesting points:

(0,3) When t=0 (at the start) the ball is at 3 m

(−0.2,0) says that −0.2 seconds BEFORE we threw the ball it was at ground level. This never happened! So our common sense says to ignore it.

(3,0) says that at 3 seconds the ball is at ground level.

Also notice that the ball goes nearly 13 meters high.

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Note: You can find exactly where the top point is!

The method is explained in Graphing Quadratic Equations, and has two steps:

Find where (along the horizontal axis) the top occurs using −b/2a:

t = −b/2a = −(−14)/(2 × 5) = 14/10 = 1.4 seconds

Then find the height using that value (1.4)

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h = −5t² + 14t + 3 = −5(1.4)² + 14 × 1.4 + 3 = 12.8 meters

So the ball reaches the highest point of 12.8 meters after 1.4 seconds.

Example: New Sports Bike

You have designed a new style of sports bicycle!

Now you want to make lots of them and sell them for profit.

Your costs are going to be:

$700,000 for manufacturing set-up costs, advertising, etc
$110 to make each bike

Based on similar bikes, you can expect sales to follow this 'Demand Curve':

Unit Sales = 70,000 − 200P

Where 'P' is the price.

For example, if you set the price:

at $0, you just give away 70,000 bikes
at $350, you won't sell any bikes at all
at $300 you might sell 70,000 − 200×300 = 10,000 bikes

So .. what is the best price? And how many should you make?

Let us make some equations!

How many you sell depends on price, so use 'P' for Price as the variable

Unit Sales = 70,000 − 200P
Sales in Dollars = Units × Price = (70,000 − 200P) × P = 70,000P − 200P²
Costs = 700,000 + 110 x (70,000 − 200P) = 700,000 + 7,700,000 − 22,000P = 8,400,000 − 22,000P
Profit = Sales-Costs = 70,000P − 200P² − (8,400,000 − 22,000P) = −200P² + 92,000P − 8,400,000

Profit = −200P² + 92,000P − 8,400,000

Yes, a Quadratic Equation. Let us solve this one by Completing the Square.

Solve: −200P² + 92,000P − 8,400,000 = 0

Step 1 Divide all terms by -200

Step 2 Move the number term to the right side of the equation:

P² – 460P = -42000

Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation:

(b/2)² = (−460/2)² = (−230)² = 52900

(P – 230)² = 10900

Step 4 Take the square root on both sides of the equation:

P – 230 = ±√10900 = ±104 (to nearest whole number)

Step 5 Subtract (-230)

Balls, Arrows, Missiles and Stones

Example: Throwing a Ball

A ball is thrown straight up, from 3 m above the ground, with a velocity of 14 m/s. When does it hit the ground?

Example: New Sports Bike

Solve: −200P2 + 92,000P − 8,400,000 = 0

Solve: −200P² + 92,000P − 8,400,000 = 0

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