Derivation of the Quadratic Formula
Let's derive the formula for solving a quadratic equation of the form:
where
, , and are real coefficients.
1. Divide Both Sides by
To simplify, divide the entire equation by :
2. Move the Constant to the Right Side
3. Complete the Square
To complete the square, add and subtract
on the left:
Group the first three terms (which form a perfect square):
4. Move Subtracted Square to the Right
5. Express Both Terms with Common Denominator
6. Substitute and Take Square Roots
Take square roots of both sides:
7. Solve for
Combine terms:
Quadratic Formula
This formula provides solutions to any quadratic equation for real coefficients , , and .
If you'd like, I can walk through a concrete example step by step!