The quadratic formula is a powerful tool for solving any quadratic equation, which is typically expressed in the standard form \( ax^2 + bx + c = 0 \). The formula is given by:

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

To effectively use the quadratic formula, it is essential to identify the coefficients \( a \), \( b \), and \( c \) from the equation. The term \( b^2 - 4ac \) is known as the discriminant, and it helps determine the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, the roots are complex.

For example, consider the equation \( x^2 + 2x - 3 = 0 \). Here, \( a = 1 \), \( b = 2 \), and \( c = -3 \). Plugging these values into the quadratic formula yields:

\( x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \)

Calculating the discriminant gives \( 2^2 - 4 \cdot 1 \cdot (-3) = 4 + 12 = 16 \). Thus, the equation simplifies to:

\( x = \frac{-2 \pm 4}{2} \)

This results in two solutions: \( x = 1 \) and \( x = -3 \).

In another example, for the equation \( x^2 - 5x = -1 \), we first rearrange it to standard form, resulting in \( x^2 - 5x + 1 = 0 \). Here, \( a = 1 \), \( b = -5 \), and \( c = 1 \). Applying the quadratic formula gives:

\( x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \)

Calculating the discriminant results in \( 25 - 4 = 21 \), leading to:

\( x = \frac{5 \pm \sqrt{21}}{2} \)

This expression indicates two solutions: \( x = \frac{5 + \sqrt{21}}{2} \) and \( x = \frac{5 - \sqrt{21}}{2} \). The quadratic formula is versatile and can be applied to any quadratic equation, making it a reliable method for finding solutions when other methods, such as factoring, are not feasible.