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The quadratic formula is an important tool for solving quadratic equations of the form ithe general form of a quadratic equation is: ax^2 + bx + c = 0 . The quadratic formula to find the roots of this equation is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. In this formula:

• a is the coefficient of x^2
• b is the coefficient of x
• c is the constant term

This formula allows us to solve for x by plugging in the values of a, b, and c. The term inside the square root , called the discriminant, influences the nature of the roots.

The discriminant is a key part of the quadratic formula, defined as: \[ b^2 - 4ac \]. The discriminant helps determine the nature of the roots of a quadratic equation without solving it.

• If \[ \Delta > 0 \], the quadratic equation has two distinct real roots.
• If \[ \Delta = 0 \], the quadratic equation has exactly one real root (a repeated root).
• If \[ \Delta < 0 \], the quadratic equation has two complex roots (no real roots).

In our example equation \[ x^2 - 2x - 4 = 0 \], substituting \( a = 1 \), \( b = -2 \), and \( c = -4 \) into the discriminant formula, we get: \[ \Delta = (-2)^2 - 4(1)(-4) = 4 + 16 = 20 \]. Since \( \Delta > 0 \), there are two distinct real roots.

Solving a quadratic equation involves several clear steps. Let's solve the equation \[ x^2 - 2x - 4 = 0 \] step by step using the quadratic formula. Step 1: Identify a, b, and c. For our equation, \( a = 1 \), \( b = -2 \), \( c = -4 \). Step 2: Calculate the discriminant. We already determined \( \Delta = 20 \). Step 3: Apply the quadratic formula: \[ x = \frac{-(-2) \pm \sqrt{20}}{2(1)} = \frac{2 \pm \sqrt{20}}{2} \]. Step 4: Simplify the equation: \[ x = \frac{2 \pm 2\sqrt{5}}{2} = 1 \pm \sqrt{5} \]. So, the solutions to the quadratic equation \( x^2 - 2x - 4 = 0 \) are \( x = 1 + \sqrt{5} \) and \( x = 1 - \sqrt{5} \). Breaking it down step by step helps ensure we correctly solve for x and understand the process.