The Quadratic Formula
For any quadratic equation ax² + bx + c = 0 with a ≠ 0, the roots are given by:
This formula was historically derived by the ancient Indian mathematician Sridharacharya by completing the square on the general equation ax² + bx + c = 0.
Applying the Formula
Solve 3x² - 5x + 2 = 0 using the Quadratic Formula
Step 1: Identify coefficients: a = 3, b = -5, c = 2.
Step 2: Calculate b² - 4ac:
Step 3: Substitute into the formula:
x = (5 ± 1) / 6
Taking positive sign: x = (5 + 1)/6 = 6/6 = 1
Taking negative sign: x = (5 - 1)/6 = 4/6 = 2/3
Roots are x = 1 and x = 2/3.
Solve x² + 4x + 5 = 0 using the Quadratic Formula
Step 1: Identify coefficients: a = 1, b = 4, c = 5.
Step 2: Calculate b² - 4ac:
Since b² - 4ac = -4 < 0, the square root of a negative number is not a real number.
Key Takeaways
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a).
Always calculate b² - 4ac first to check if real roots exist.
Be careful with signs, especially when b or c is negative!