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The Quadratic Formula is a powerful tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula gives us a way to find the roots of these equations by substituting the coefficients directly into a standard formula:

• a: Coefficient of \(x^2\)
• b: Coefficient of \(x\)
• c: Constant term

The formula itself is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows us to find the values of \(x\) that make the equation true. Its use is straightforward once you substitute the coefficients correctly. For our specific equation \(x^2 - 4x + 2 = 0\), we identify \(a = 1\), \(b = -4\), and \(c = 2\) and plug these into the formula to determine the roots.

The discriminant is a critical part of the Quadratic Formula that helps us determine the nature of the roots. It is represented by the expression under the square root sign:\(b^2 - 4ac\).The value of the discriminant reveals insightful information:

• Positive Discriminant: If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• Zero Discriminant: If \(b^2 - 4ac = 0\), there is exactly one real root, which means the roots are repeated.
• Negative Discriminant: If \(b^2 - 4ac < 0\), the equation has no real roots, only complex ones.

In our example, the discriminant is calculated as \(16 - 8 = 8\). Since it is positive, we know there are two distinct real solutions for the equation.

The solutions or roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Using the Quadratic Formula, after calculating and simplifying, we determine our roots. For example, with the quadratic equation \(x^2 - 4x + 2 = 0\), the formula gives us:\[x = \frac{-(-4) \pm \sqrt{8}}{2}\]The simplification steps further result in:\[x = 2 \pm \sqrt{2}\]These computations give us two solutions: \(x = 2 + \sqrt{2}\) and \(x = 2 - \sqrt{2}\). These solutions are the roots of the quadratic equation; they mark the points where the parabola represented by the equation will intersect the x-axis.Understanding this process is essential for solving quadratic equations and analyzing their graphs.