NapkinCalc

Algebra 2

Quadratics & the discriminant

A quadratic ax² + bx + c = 0 has its whole personality in one number, the discriminant b² − 4ac:

  • positive → two real roots (the parabola crosses the axis twice)
  • zero → one root (it just touches)
  • negative → no real roots (it never reaches the axis)
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
r2=−bq−D2⋅aqr_{2} = \frac{-b_{q} - \sqrt{D}}{2 \cdot a_{q}} = −2-2 second root
xv=−bq2⋅aqx_{v} = \frac{-b_{q}}{2 \cdot a_{q}} = 11 the vertex sits exactly between the roots
✓ 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
a_q * x^2 + b_q * x + c_q
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the parabola — roots where it crosses zero, vertex at x_v