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The quadratic formula is a powerful tool in algebra for solving equations of the form \( ax^2 + bx + c = 0 \). It provides us a way to find the values of \( x \) that satisfy the equation. The formula itself is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula directly gives us the roots of any quadratic equation by plugging in the values of \( a \), \( b \), and \( c \). The term under the square root (\( b^2 - 4ac \)) is called the discriminant, and it determines the nature of the roots:

• If the discriminant is positive, the equation has two distinct real solutions.
• If the discriminant is zero, there is exactly one (repeated) real solution.
• If the discriminant is negative, there are no real solutions, but rather two complex solutions.

Thus, with this formula, not only do we find the solutions, but we also understand how many there are and their nature.

In algebra, real solutions refer to the values that satisfy the equation and lie within the set of real numbers. Real solutions play a crucial role because they provide us with the actual value outcomes we seek, especially in practical situations.When solving polynomial equations like the one given \( x^4 + 3x^2 - 2 = 0 \), the key is to determine the nature of the solutions based on the derived quadratic equation from substitution. After substitution, our polynomial becomes a simpler quadratic form \( y^2 + 3y - 2 = 0 \), where \( y = x^2 \).Applying the quadratic formula to this simplified form helps us find the potential values of \( y \). However, not all values of \( y \) translate to real \( x \) solutions. This is because \( x^2 = y \), meaning \( y \) must be non-negative for \( x \) to have real solutions. We discard any negative \( y \) value as it cannot yield real \( x \) solutions. Hence, real solutions are derived from valid, non-negative \( y \) values retrieved from the quadratic equation.

Taking square roots is a common step when deriving solutions in many algebraic problems, especially when solving equations involving squares. When you solve a quadratic equation and determine \( x^2 = \text{{some non-negative value}} \), taking the square root provides both the positive and negative solutions of \( x \).For instance, in our problem, once the value \( y = x^2 = \frac{-3 + \sqrt{17}}{2} \) is computed, we proceed by taking the square root of both sides. This gives:\[ x = \pm \sqrt{\frac{-3 + \sqrt{17}}{2}} \]The square root operation here is crucial because it reflects the concept that any real number squared will result in a non-negative number. By taking the square root, we retrieve both possible values for \( x \), thus ensuring that we account for all possible real solutions.In calculations with real numbers, using the square root efficiently helps us transition from squared values to the actual solutions we can potentially observe and utilize.