The Quadratic Formula

The quadratic formula offers a powerful method for solving quadratic equations when other techniques like factoring prove challenging. This page introduces the formula and provides several examples of its application.

Definition: The quadratic formula states that for a quadratic equation in the form ax² + bx + c = 0, the solutions are given by x = (-b ± √(b² - 4ac)) / (2a).

Highlight: The quadratic formula works for all quadratic equations, even those with complex roots.

The page walks through multiple examples, demonstrating how to:

1. Identify the coefficients a, b, and c in the equation
2. Substitute these values into the quadratic formula
3. Simplify and solve for x

Example: For x² + 8x = -15, the page shows how to rearrange it to standard form (x² + 8x + 15 = 0), identify a=1, b=8, c=15, and then apply the formula to find solutions x = -5 and x = -3.

Additional examples include:

• Solving (2x+7)² = 25 by first simplifying to standard form
• Writing a quadratic equation given its roots (-2 and -3)
• Solving x² + 5x - 10 = 0, demonstrating a case where the solution involves a square root that cannot be simplified further

Vocabulary: Standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

The page concludes with a complex example: (2x+3)(x+4) = 1, showing how to expand and rearrange before applying the quadratic formula.

This comprehensive guide provides students with the tools to solve quadratic equations using the quadratic formula across a variety of problem types, reinforcing the formula's versatility and importance in algebra.