math class 678 4-in-1 skill sheets
The Discriminant and Number of Solutions, Algebra 1 4-in-1 Skill Sheet (HSA.REI.B.4b)
HSA.REI.B.4b Algebra 1 Unit 13: Quadratic Equations

The Discriminant and Number of Solutions

I can use the discriminant to say how many real solutions a quadratic has, before solving it.

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The problem this sheet solves

The discriminant is what goes UNDER the plus-or-minus sign, and adding zero and subtracting zero land in the same place. That is one solution, and it is the most interesting of the three cases rather than the empty one. The opposite error runs alongside it: a discriminant that is positive but not a perfect square gets read as no solutions because nothing factors. Teach the position, not the rule — a student who can say where the discriminant sits has already answered why zero gives one. The reference figure is three parabolas in the SAME window, so the only difference between the pictures is the thing being taught: one cuts the axis, one touches it, one never reaches it. This is the only tool in the unit that answers a question without solving.

The key rule

D = b2 − 4ac. Positive gives two real solutions, zero gives one, negative gives none.

Zero is one, not none.

Watch out

The mistake: A student computes D = 0 and answers no solutions.

The fix: Zero goes under the ± sign, and adding zero and subtracting zero land in the same place. That is ONE solution: x = 3. The parabola touches the axis there.

What is included

  • Reference page — student-friendly definitions, a worked example, a Watch Out error pair, and the key rule
  • Practice page — six problems, building from foundational to multi-step
  • Apply page — a word problem, a reasoning prompt, and an error-analysis task
  • Assess — a half-page exit ticket, two to a sheet, with a score box
  • Full answer key — every item worked, plus a grading guide that says what each wrong answer tells you
  • Ink-saver edition of all 7 pages
  • 14-slide teacher deck (PowerPoint) with presenter notes on every slide

Standard codes refer to the Common Core State Standards for Mathematics. Math Class 678 is an independent publisher, not affiliated with, sponsored by, or endorsed by NGA Center or CCSSO. Every product is sold on Teachers Pay Teachers.