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A Quadratic Equation looks like this:

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Quadratic equations pop up in many real world situations!

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Fraction Word Problems, The first example is a one-step word problem, The second example shows how blocks can be used to help illustrate the problem, The third example is a two-step word problem, bar modeling method in Singapore Math, Word Problem on Subtracting Fractions From Whole Numbers, with video lessons, examples and step-by-step solutions.

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)

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.

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)

• 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:

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

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

Step 5 Subtract (-230) from both sides (in other words, add 230):

What does that tell us? It says that the profit is ZERO when the Price is $126 or $334

But we want to know the maximum profit, don't we?

It is exactly half way in-between! At $230

And here is the graph:

Profit = −200P² + 92,000P − 8,400,000

The best sale price is $230, and you can expect:

• Unit Sales = 70,000 − 200 x 230 = 24,000
• Sales in Dollars = $230 x 24,000 = $5,520,000
• Costs = 700,000 + $110 x 24,000 = $3,340,000
• Profit = $5,520,000 − $3,340,000 = $2,180,000

A very profitable venture.

Example: Small Steel Frame

Your company is going to make frames as part of a new product they are launching.

The frame will be cut out of a piece of steel, and to keep the weight down, the final area should be 28 cm²

The inside of the frame has to be 11 cm by 6 cm

What should the width x of the metal be?

Area of steel before cutting:

Area of steel after cutting out the 11 × 6 middle:

Let us solve this one graphically!

Here is the graph of 4x² + 34x :

The desired area of 28 is shown as a horizontal line.

The area equals 28 cm² when:

x is about −9.3 or 0.8

The negative value of x make no sense, so the answer is:

x = 0.8 cm (approx.)

Example: River Cruise

A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey?

There are two speeds to think about: the speed the boat makes in the water, and the speed relative to the land:

• Let x = the boat's speed in the water (km/h)
• Let v = the speed relative to the land (km/h)

Because the river flows downstream at 2 km/h:

• when going upstream, v = x−2 (its speed is reduced by 2 km/h)
• when going downstream, v = x+2 (its speed is increased by 2 km/h)

We can turn those speeds into times using:

time = distance / speed

(to travel 8 km at 4 km/h takes 8/4 = 2 hours, right?)

And we know the total time is 3 hours:

total time = time upstream + time downstream = 3 hours

Put all that together:

total time = 15/(x−2) + 15/(x+2) = 3 hours

Now we use our algebra skills to solve for 'x'.

First, get rid of the fractions by multiplying through by (x-2)(x+2):

3(x-2)(x+2) = 15(x+2) + 15(x-2)

Expand everything:

3(x²−4) = 15x+30 + 15x−30

Bring everything to the left and simplify:

3x² − 30x − 12 = 0

It is a Quadratic Equation! Let us solve it using the Quadratic Formula:

Where a, b and c are from the
Quadratic Equation in 'Standard Form': ax² + bx + c = 0

Solve 3x² - 30x - 12 = 0

Answer: x = −0.39 or 10.39 (to 2 decimal places)

x = −0.39 makes no sense for this real world question, but x = 10.39 is just perfect!

Answer: Boat's Speed = 10.39 km/h (to 2 decimal places)

And so the upstream journey = 15 / (10.39−2) = 1.79 hours = 1 hour 47min

And the downstream journey = 15 / (10.39+2) = 1.21 hours = 1 hour 13min

Example: Resistors In Parallel

Two resistors are in parallel, like in this diagram:

The total resistance has been measured at 2 Ohms, and one of the resistors is known to be 3 ohms more than the other.

What are the values of the two resistors?

The formula to work out total resistance 'R_T' is:

1R_T = 1R₁ + 1R₂

In this case, we have R_T = 2 and R₂ = R₁ + 3

12 = 1R₁ + 1R₁+3

To get rid of the fractions we can multiply all terms by 2R₁(R₁ + 3) and then simplify:

Yes! A Quadratic Equation !

Let us solve it using our Quadratic Equation Solver.

• Enter 1, −1 and −6
• And you should get the answers −2 and 3

R₁ cannot be negative, so R₁ = 3 Ohms is the answer.

The two resistors are 3 ohms and 6 ohms.

Others

Quadratic Equations are useful in many other areas:

For a parabolic mirror, a reflecting telescope or a satellite dish, the shape is defined by a quadratic equation.

Quadratic equations are also needed when studying lenses and curved mirrors.

And many questions involving time, distance and speed need quadratic equations.

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#1 OhmConnect

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Here’s a money making app that doesn’t make you watch boring ads or waste any of your time.

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#2- Ibotta [US ONLY]

Ibotta is an app that pays you for shopping. If you buy your groceries or beauty products online, you can choose from a select number of items listed in the app.

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Download Ibotta - iOS | Android | Web

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Other Apps that Pay to Shop:

#3 - Receipt Hog:iOS|Android|Web

#4 - ShopKick:iOS|Android|Web

#5 - Mobee App:iOS|Android|Web

#6 - CashPirate

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If you enjoy playing games, completing surveys, or watching videos, CashPirate is a money making app you may want to check out. You can choose which money making options you like best and earn extra cash whenever you have the time.

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For every 1000 coins, you earn $1. You must earn at least 2500 coins before you’ll be able to cash out and send money to your PayPal account. You can expect to earn 50 coins for downloading an app.

Download CashPirate - iOS | Android | Web

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Money Making Apps Similar to CashPirate:

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#7 - AppKarma: iOS (Invitation Only) |Android|Web

#8 - Make Money:iOS|Android

#9 - Panel App:iOS|Android|Web

#10 - FeaturePoints:iOS|Android|Web

#11 - SwagBucks

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If you enjoy taking surveys, you can earn as much money as you can answer with SwagBucks.

SwagBucks is one of the top known money making apps that actually pays users through PayPal. They have endless surveys to take and you can also earn money by playing games, shopping online, weekly contests and more.

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Survey Taking Apps That Pay Similar to SwagBucks:

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#12 - GlobalTestMarket:Web

#13 - PaidViewPoint:Web

#14 - InstaGC:Web

#15 - Toluna:iOS|Android|Web

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#16 - CashCrate:iOS|Android|Web

#17 - SlideJoy

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Get paid just for locking your screen. SlideJoy is an app that pays you to place ads or news on your screen. It’s one of the most simple ways to get paid and you can even choose to see trending news tailored to your interest.

Download SlideJoy - Android | Web

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Apps that Pay like SlideJoy:

#18 - Fronto:Android|Web

#19 - AppTrailers

AppTrailers lets you get paid for watching videos and you can put real money into your PayPal account.

Whenever you watch videos Google play or any other web based browser, you’ll get rewarded. You’ll be watching ads and you may you be participating in market research from Nielson’s TV Rating. You can learn more about this in their privacy policy.

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Download AppTrailers - iOS | Android | Web

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Get Paid with Apps Similar to AppTrailers:

#20 - Viggle: iOS | Android | Web

#21 - SwagBucks Watch (TV): Android | Web

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