NapkinCalc

Algebra 2A — Functions & Quadratics

Quadratics & the discriminant

continues from lesson 1 — values defined earlier in the course stay live here

ELI5: a quadratic ax² + bx + c = 0 graphs as a parabola (a U or ∩). Its entire personality hides in one number, the discriminant b² − 4ac:

  • positive → two real roots (crosses the axis twice)
  • zero → one root (just kisses the axis)
  • negative → no real roots (never reaches it)
aq=1a_{q} = 1
bq=−2b_{q} = -2
cq=−8c_{q} = -8
D=bq2−4⋅aq⋅cqD = b_{q}^{2} - 4 \cdot a_{q} \cdot c_{q} = 3636 the discriminant — try values that make it 0 or negative
r1=−bq+D2⋅aqr_{1} = \frac{-b_{q} + \sqrt{D}}{2 \cdot a_{q}} = 44 first root via the quadratic formula
r2=−bq−D2⋅aqr_{2} = \frac{-b_{q} - \sqrt{D}}{2 \cdot a_{q}} = −2-2 second root
✓ pass abs(aq⋅r12+bq⋅r1+cq)<10−9\mathrm{abs}\left(a_{q} \cdot r_{1}^{2} + b_{q} \cdot r_{1} + c_{q}\right) < 10^{-9} r₁ really is a root
rootsq=solve(aq⋅x2+bq⋅x+cq,x)roots_{q} = \mathrm{solve}\left(a_{q} \cdot x^{2} + b_{q} \cdot x + c_{q}, x\right) = [4,−2][4, -2] or skip the formula — solve() returns BOTH roots
✓ pass max(rootsq)==max(r1,r2)andmin(rootsq)==min(r1,r2)max(roots_{q}) == max(r_{1}, r_{2}) and min(roots_{q}) == min(r_{1}, r_{2}) same two roots the formula gave
a_q * x^2 + b_q * x + c_q
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the parabola — roots where it crosses zero

Real-world hook: parabolas are anything thrown (a ball, a jet of water), the shape of satellite dishes and headlights, and the profit curve whose peak a business hunts for.