Mathematics · Grades 11–12
Quadratic equations · Formula sheet
Rules, conditions and small examples in one printable reference.
Remember this
Evaluate a quadratic
Square the input before multiplying by its coefficient.
f(x) = ax² + bx + c
6 × (-5)² + 1 × (-5) + 7 = 152
The discriminant
For real coefficients with a ≠ 0: positive Δ gives two real roots; zero gives one repeated root; negative gives no real roots.
Δ = b² − 4ac
15² − 4 × 1 × 5 = 205
Roots from factors
Set each factor equal to zero and check the original equation.
uv = 0 → u = 0 or v = 0
x = 24 or x = 26; 24 < 26
Axis of symmetry
The vertex lies on this vertical axis for y = ax² + bx + c.
x = −b/(2a), a ≠ 0
k = −33/(2 × 1) = -33/2
Expand a binomial square
The middle term is twice the product, not zero.
(x + a)² = x² + 2ax + a²
2 × 6 × 24 = 288
Complete the square
Halve the coefficient of x, then square it.
x² + 2hx + h² = (x + h)²
(68 ÷ 2)² = 1156
Further reading and review references
These are reference links, not copied exercises, institutional endorsement or proof of independent review.
Learning preview · teacher review required. Grade bands are approximate, not official curriculum alignment.