Equations with x² (ax² + bx + c = 0): the quadratic formula, incomplete ones and when there is no real solution. Every step, the check and the parabola graph.
Equation
Write it with an = sign. Without =, it is taken as equal to 0. x², x^2, 0.5x, 3(x − 2) and fractions like x/2 all work.
Examples:
Solution
x = 2 · x = 3
y = x² − 5x + 6y = 0
Points
Solution:x = 2
Solution:x = 3
Type
Quadratic
Solutions
2
Solving
x² − 5x + 6 = 0
Quadratic formula with a = 1, b = −5, c = 6: x = (−b ± √(b² − 4ac)) / 2a
Discriminant: b² − 4ac = (−5)² − 4·1·6 = 1
x = (5 ± √1) / 2 → x₁ = 2 and x₂ = 3
Check
Substitute each solution into both sides: they must match.
With x = 2: left = 0, right = 0 ✓
With x = 3: left = 0, right = 0 ✓
On the graph
Each side is a function: y = x² − 5x + 6 (blue) and y = 0 (pink). The solutions are the x values where the two graphs cross.