Extracting the square root from both parts of an equation

If you extract the square root from both parts of the equation, you get an equation equal to the original one.

Consider the following equation:

x^{2} = 16

This is the simplest quadratic equation that has two roots: 4 and -4. We solved such an equation using the definition of a square root.

According to the definition of a square root, b is the square root of a if b^{2} = a and is denoted as b = âˆša.

Then in the case of x^{2} = 16, we can write that x = âˆš16, whence x = Â±4.

Now solve this quadratic equation by extracting the square root from both parts of the equation.

"Wrap" both parts of the equation x^{2} = 16 to the square root:

Now recall one of the properties of the square root, which states that the square root of the square of a number is equal to the modulus of that number

Then on the left side of our equation we get the modulus of x, and on the right side we get the number 4

We obtained the simplest equation with a modulus. It has two roots: 4 and -4. Write down this solution as a set of equations:

Checking:

It is the arithmetic square root that must be extracted from the right side of the equation x^{2} = 16. Earlier we said that the square root has two values: positive and negative. That is:

But in this case we are interested in the non-negative value of 4 (which is called the arithmetic square root). Because if we extract the second root (negative -4), we get the equation |x|= -4, which has no solutions.

Example 2. Solve the equation 3x^{2} = 12

Solution

Divide both parts by 3

Extract the square root of both parts of the resulting equation:

We obtained the simplest equation with a module. Solve it by reducing it to a set:

Answer: 2 and -2.

Example 3. Solve the equation (x + 2)^{2} = 25

Solution

Extract the square root of both parts of the resulting equation:

Solve the resulting equation with a module:

Answer: 3 and -7.

Example 4. Solve the equation x^{2} - 10 = 39

Solution

Move -10 to the right side by changing the sign:

Extract the square root of both parts of the resulting equation:

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