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The quadratic formula is a reliable tool used to find solutions of any quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. It is particularly useful when factorizing isn't straightforward. The formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

To utilize this formula correctly, first identify the coefficients in your quadratic equation. Then, simply plug these values into the formula. It is important to ensure all arithmetic operations, especially inside the square root, are precise, as small errors can lead to incorrect answers. Remember that the "\( \pm \)" symbol indicates there are potentially two solutions for \( x \).
In our example, with coefficients \( a = 4 \), \( b = -6 \), and \( c = -4 \), substituting into the formula helps find possible values of \( x \) that satisfy the original equation.

The discriminant is a key component of the quadratic formula and plays an essential role in determining the nature and number of solutions to a quadratic equation.

• It is defined as \( b^2 - 4ac \) within the quadratic formula.
• The value of the discriminant reveals important information about the roots of the equation:
• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root (a repeated root).
• If \( b^2 - 4ac < 0 \), no real roots exist (the roots are complex numbers).

For the given problem, the calculated discriminant is 100, which means two distinct real solutions exist. This aligns perfectly with the quadratic equation giving two results, out of which only the valid root fulfills the original radical equation.

Solving radical equations involves isolating the radical expression and eliminating it through squaring. Let's break it down:

• First, ensure the radical term, such as \( \sqrt{8-2x} \), is isolated. This means the equation should be set so that the square root is the only term on one side.
• Next, to remove the square root, square both sides. Remember, whatever operation you perform on one side of the equation must also be done on the other.
• Squaring eliminates the radical, providing a polynomial equation that can be solved further, often requiring use of the quadratic formula.

In our example, once the radical is squared away, we are left with a quadratic equation which we solve using the quadratic formula. Don't forget to check each solution by substituting back into the original radical equation because squaring can introduce extraneous solutions.