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The discriminant worksheets Grades 10–11

Use b² − 4ac to tell how many real roots a quadratic has.

The front of the free the discriminant worksheet
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Free · Worked example · 18 problems · Answer key · US Letter

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How to do it

b² − 4ac more than 0: two real roots. Equal to 0: one real root. Less than 0: two complex roots.

−2x2−8x−8=0b2−4ac: 0, one real root-2x^{2} - 8x - 8 = 0\quad b^{2} - 4\text{ac: }0,\ \text{one real root}
(−8)2−4(−2)(−8)=0(-8)^{2} - 4(-2)(-8) = 0b² − 4ac
0 is equal to 0: one real root0\ \text{is equal to }0:\ \text{one real root}

Example problems

  1. 4x2+4x+1=0b2−4ac: 4x^{2} + 4x + 1 = 0\quad b^{2} - 4\text{ac: }
  2. −3x2+2x−2=0b2−4ac: -3x^{2} + 2x - 2 = 0\quad b^{2} - 4\text{ac: }
  3. −4x2+8x−4=0b2−4ac: -4x^{2} + 8x - 4 = 0\quad b^{2} - 4\text{ac: }
  4. 2x2+4x+2=0b2−4ac: 2x^{2} + 4x + 2 = 0\quad b^{2} - 4\text{ac: }

Quadratic and polynomial functions

  1. Vertex form of a quadratic
  2. Finding the vertex from standard form
  3. The discriminant
  4. Quadratic equations with complex roots
  5. Polynomial long division
  6. Synthetic division
  7. The remainder theorem
  8. The factor theorem
  9. The rational root theorem