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:
- Identify the coefficients a, b, and c in the equation
- Substitute these values into the quadratic formula
- 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+7² = 25 by first simplifying to standard form
- Writing a quadratic equation given its roots −2and−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+3x+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.
