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Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. These equations are called 'quadratic' because the highest exponent of the variable \( x \) is 2.

To solve a quadratic equation, several methods can be used such as:

• Factoring
• Using the quadratic formula \( x = \frac{-b \text{±} \text{√}(b^2-4ac)}{2a} \)
• Completing the square
• Graphing

In the exercise above, we solve the quadratic equation \( 4x^2 = 20 \). The solution process involves simplifying the equation by dividing both sides by 4, reducing it to \( x^2 = 5 \). From here, we can easily solve for \( x \) by taking the square root of both sides.

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \).

In mathematical notation, the square root of a number \( y \) is written as \( \text{√}y \). Every non-negative real number has two square roots: a positive and a negative one, represented as \( \text{±} \text{√}y \).

In our exercise, after reducing \( 4x^2 = 20 \) to \( x^2 = 5 \), we then take the square root of both sides. This gives us two possible solutions for \( x \):

• \( x = \text{√}5 \)
• \( x = -\text{√}5 \)

This is because both \( (\text{√}5)^2 \) and \( (-\text{√}5)^2 \) equal 5.
Understanding square roots is essential in solving quadratic equations as they help in determining the values of the variable.

To solve for a variable, it's often necessary to isolate it on one side of the equation. 'Isolation of variable' means manipulating the equation to get the variable alone.

In our example, the quadratic equation \( 4x^2 = 20 \) requires isolating \( x^2 \). We do this by dividing both sides of the equation by 4, resulting in \( x^2 = 5 \).

Steps for isolating the variable can include:

• Adding or subtracting terms on both sides
• Multiplying or dividing both sides by a constant
• Using inverse operations (like square root) to undo functions applied to the variable

In this case, once \( x^2 \) is isolated, taking the square root of both sides directly gives us the solutions for \( x \). Isolating the variable is a key step in making complex equations simpler and solvable.