courses › Algebra

Roots, the discriminant, the vertex, and the formula.

Pen and paper is fine · a calculator helps on some problems why?

Learn first (about 3 minutes)

Opens at level 28.

Sign in to start

Builds on: Polynomials (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Roots by factoring par 25 s not tried yet

  worked example

  Solve 3x² − 21x = 0. Enter both roots, smaller first.

  Answer: (0, 7)

  1. Divide by 3: x² − 7x = 0.
  2. Factor: x(x − 7) = 0.
  3. So x = 7 or x = 0. Smaller first: 0, 7.

• Discriminant par 16 s not tried yet

  worked example

  What is the discriminant of 2x² − 4x + 10?

  Answer: -64

  1. b² − 4ac = (−4)² − 4 × 2 × 10 = 16 − 80 = −64.
  2. It is negative, so there are no real roots.

• Vertex par 25 s not tried yet

  worked example

  What is the vertex of y = x² + 10x + 9? Enter x, y.

  Answer: (-5, -16)

  1. x = −b ÷ (2a) = −10 ÷ 2 = −5.
  2. y = (−5)² + 10 × (−5) + 9 = 25 − 50 + 9 = −16.

• Quadratic formula par 45 s not tried yet

  worked example

  Solve x² + 2x − 7 = 0. Give the larger root, rounded to 2 decimal places.

  Answer: 1.83

  1. x = (−b ± √(b² − 4ac)) ÷ (2a) = (−2 ± √(4 − (−28))) ÷ 2 = (−2 ± √32) ÷ 2.
  2. √32 ≈ 5.6569, so the larger root is (−2 + 5.6569) ÷ 2 ≈ 1.8284.
  3. Rounded to 2 decimal places: 1.83.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

your rounds

No rounds yet.