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10.5: Square Root Applications

Difficulty Level: At Grade Created by: CK-12
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Square Root Applications 

We can use the methods we’ve learned so far in this section to find approximate solutions to quadratic equations, when taking the square root doesn’t give an exact answer.

Solving Quadratic Equations 

1. Solve the following quadratic equations.


Answer: and


Answer: and

2. Solve the following quadratic equations.


Answer: and


Answer: and

Solve Applications Using Quadratic Functions and Square Roots

Quadratic equations are needed to solve many real-world problems. In this section, we’ll examine problems about objects falling under the influence of gravity. When objects are dropped from a height, they have no initial velocity; the force that makes them move towards the ground is due to gravity. The acceleration of gravity on earth is given by the equation

The negative sign indicates a downward direction. We can assume that gravity is constant for the problems we’ll be examining, because we will be staying close to the surface of the earth. The acceleration of gravity decreases as an object moves very far from the earth. It is also different on other celestial bodies such as the moon.

The equation that shows the height of an object in free fall is

The term represents the initial height of the object, is time, and is the constant representing the force of gravity. You then plug in one of the two values for above, depending on whether you want the answer in feet or meters. Thus the equation works out to if you want the height in meters, and if you want it in feet.

Real-World Application: Object Falling 

1. How long does it take a ball to fall from a roof to the ground 25 feet below?

Since only positive time makes sense in this case, it takes the ball 1.25 seconds to fall to the ground.

2. A rock is dropped from the top of a cliff and strikes the ground 7.2 seconds later. How high is the cliff in meters?

The cliff is 254 meters high.


Example 1

Victor throws an apple out of a window on the floor which is 120 feet above ground. One second later Juan throws an orange out of a floor window which is 72 feet above the ground. Which fruit reaches the ground first, and how much faster does it get there?

Let’s find the time of flight for each piece of fruit.



The orange was thrown one second later, so add 1 second to the time of the orange:

The apple hits the ground first. It gets there 0.38 seconds faster than the orange.


Solve the following quadratic equations.

  1. Susan drops her camera in the river from a bridge that is 400 feet high. How long is it before she hears the splash?
  2. It takes a rock 5.3 seconds to splash in the water when it is dropped from the top of a cliff. How high is the cliff in meters?
  3. Nisha drops a rock from the roof of a building 50 feet high. Ashaan drops a quarter from the top story window, 40 feet high, exactly half a second after Nisha drops the rock. Which hits the ground first?

Review (Answers)

To view the Review answers, open this PDF file and look for section 10.5. 

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quadratic function A quadratic function is a function that can be written in the form f(x)=ax^2 + bx + c, where a, b, and c are real constants and a\ne 0.
Square Root The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9.

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Difficulty Level:
At Grade
Date Created:
Oct 01, 2012
Last Modified:
Apr 11, 2016
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