Activity
A whole number that has two factors that are equal to each other is called a perfect square. Those equal factors that multiply to be the number are called the square root of a number. For instance, is a square root of because .
SQUARE ROOT PROPERTY
If , then
Each of these equations has two solutions. Use your knowledge of square roots and the square root property. What are the solutions? Explain or show your reasoning.
Enter the solutions and your reasoning.
Compare your answer:
and
If , then . Using the square root property, . Therefore, and .
Enter the solutions and your reasoning.
Compare your answer:
and
Divide each side of the equation by 3. Using the square root property, . Therefore, and .
Enter the solutions and your reasoning.
Compare your answer:
and
Add 20 to each side of the equation. Then multiply each side of the equation by 5. Using the square root property, . Therefore, and .
Enter the solutions and your reasoning.
Compare your answer:
and
Add 7 to each side of the equation. Divide each side of the equation by 2. Using the square root property, . Therefore, and .
Are you ready for more?
Extending Your Thinking
How many solutions does the equation have? What are the solutions?
Enter the number of solutions and the solutions.
Compare your answer:
Three solutions: , , and .
How many solutions does the equation have? What are the solutions?
Enter the number of solutions and the solutions.
Compare your answer:
Two solutions: and .
Write a new equation that has 10 solutions.
Enter an equation.
Compare your answer:
There are many different possible answers. For example:
Self Check
Additional Resources
Using the Square Root Property to Solve Quadratic Equations
Let’s look again at the square root property.
SQUARE ROOT PROPERTY
If , then
Notice that the square root property gives two solutions to an equation of the form , the principal square root of and its opposite. We could also write the solution as . We read this as equals positive and negative the square root of .
Example 1
Let’s solve using the square root property.
So, and .
If the number in the radical is not a perfect square, leave that part inside the radical. Let’s solve another example.
Example 2
Solve .
Step 1 - Isolate the quadratic term and make its coefficient 1.
To do this, add 50 to both sides of the equation to get by itself.
Step 2 - Use the square root property.
Remember to write the symbol.
Step 3 - Simplify the radical by pulling out the factors that are perfect squares.
,
Step 4 - Check the solutions.
The steps for solving a quadratic equation using the square root property are listed here.
How to solve a quadratic equation using the square root property.
Step 1 - Isolate the quadratic term and make its coefficient 1.
Step 2 - Use the square root property.
Step 3 - Simplify the radical.
Step 4 - Check the solutions.
Before using the square root property in Step 2, the coefficient of the variable term must equal 1. In the next example, we must divide both sides of the equation by the coefficient 3 before using the square root property.
Example 3
Solve .
Step 1 - Isolate the quadratic term and make its coefficient 1.
When the quadratic term is isolated, divide by 3 to make its coefficient 1. Simplify.
Step 2 - Use the square root property.
Step 3 - Simplify the radical.
Rewrite to show two solutions.
,
Step 4 - Check the solutions.
Try it
Try It: Using the Square Root Property to Solve Quadratic Equations
Solve the following quadratic equations using the square root property.
1.
2.
Here is how to solve these quadratic equations using the square root property: