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IB DP Maths AA · Topic 2 Functions · SL 2.6, 2.7, 2.10 · ~6 min read

Quadratics and the discriminant

The three forms of a quadratic, completing the square, the vertex and axis of symmetry, solving by factorising and the formula, the discriminant and the number of roots, quadratic inequalities and hidden quadratics.

Standard, factorised & vertex formCompleting the squareAxis of symmetryQuadratic formulaDiscriminantInequalitiesHidden quadratics
RSby Riyaz Sir
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No tag = SL and HLHL ONLY HL students onlyFORMULA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • A quadratic y = ax2 + bx + c has three useful forms: standard (y-intercept c), factorised a(x − p)(x − q) (roots p and q) and vertex a(x − h)2 + k (vertex (h, k)).
  • The axis of symmetry is x = −b/2a. If a > 0 the graph is U-shaped (minimum); if a < 0 it is ∩-shaped (maximum).
  • The discriminant Δ = b2 − 4ac tells you how many real roots: two (Δ > 0), one repeated (Δ = 0) or none (Δ < 0).
  • Hidden quadratics such as x4 − 5x2 + 4 = 0 or 22x − 6(2x) + 8 = 0 become ordinary quadratics with a substitution.
  • This note is the same at SL and HL, and unchanged for 2029.

By the end you can

  • Convert between standard, factorised and vertex forms, including completing the square.
  • Find the vertex, axis of symmetry and intercepts, and sketch the graph.
  • Solve quadratic equations by factorising, completing the square and the formula.
  • Use the discriminant to decide the number of roots, including in problems with a parameter.
  • Solve quadratic inequalities and equations that reduce to quadratics.

1Forms of a quadratic

xyvertex (3, −4)(1, 0)(5, 0)(0, 5)axis x = 3y = x² − 6x + 5= (x − 1)(x − 5)= (x − 3)² − 4
One quadratic, three forms. Each form shows a different feature straight away.
x = −b2a FORMULA BOOKLET
axis of symmetry
Worked example

Write y = x2 − 6x + 5 in vertex form and state the vertex.

  1. x2 − 6x + 5 = (x − 3)2 − 9 + 5 M1
  2. = (x − 3)2 − 4 A1
  3. vertex (3, −4), a minimum A1
Worked example

Write y = 2x2 + 12x + 7 in the form a(x − h)2 + k.

  1. = 2(x2 + 6x) + 7 = 2[(x + 3)2 − 9] + 7 M1
  2. = 2(x + 3)2 − 11 A1
Common mistakeForgetting to multiply the "− 9" by the 2 outside the bracket. Factor out a first, complete the square inside, then multiply back.

2Solving quadratic equations

x = −b ± √(b2 − 4ac)2a FORMULA BOOKLET
the quadratic formula
Worked example

Solve 3x2 − 5x − 1 = 0, giving exact answers.

  1. x = 5 ± √(25 + 12)6 M1
  2. x = 5 ± √376 A1
  • Which method? Factorise if it is obvious; otherwise the formula. Completing the square gives exact roots AND the vertex at the same time.

3The discriminant

xΔ > 0two real rootsxΔ = 0one repeated rootxΔ < 0no real rootsΔ = b² − 4ac decides how many times the graph meets the x-axis
Shift the same parabola up and the number of real roots drops from two to one to none.
Δ = b2 − 4ac FORMULA BOOKLET
Δ > 0 two real roots; Δ = 0 one repeated root; Δ < 0 no real roots
Worked example

Find k so that x2 + kx + 9 = 0 has equal roots.

  1. Δ = k2 − 36 = 0 M1
  2. k = ±6 A1
Worked example

Find m so that the line y = mx is tangent to y = x2 + 3.

  1. x2 − mx + 3 = 0 must have one repeated root M1
  2. Δ = m2 − 12 = 0 A1
  3. m = ±2√3 A1
Worked example

For which k does kx2 + 4x + k = 0 have two distinct real roots?

  1. Δ = 16 − 4k2 > 0 ⇒ k2 < 4 M1
  2. −2 < k < 2, and k ≠ 0 (otherwise it is not a quadratic) A1R1
Exam tip"Line meets curve" questions: set them equal, rearrange to = 0, then use Δ. Tangent ⇔ Δ = 0; meets twice ⇔ Δ > 0; misses ⇔ Δ < 0.

4Quadratic inequalities

Worked example

Solve x2 − x − 6 < 0.

  1. (x − 3)(x + 2) < 0 M1
  2. the graph is below the x-axis between its roots R1
  3. −2 < x < 3 A1
Common mistakeWriting "x < 3 and x < −2". Sketch the parabola: for "< 0" the answer is between the roots; for "> 0" it is outside them (x < −2 or x > 3).

5Equations that reduce to quadratics

Worked example

Solve x4 − 5x2 + 4 = 0.

  1. let u = x2: u2 − 5u + 4 = 0 ⇒ u = 1 or 4 M1A1
  2. x = ±1, ±2 A1
Worked example

Solve 22x − 6(2x) + 8 = 0.

  1. let u = 2x: u2 − 6u + 8 = 0 ⇒ u = 2 or 4 M1
  2. 2x = 2 or 4, so x = 1 or 2 A1

6Quadratic models

Worked example

A ball's height is h = 20t − 5t2 (metres, t in seconds). Find the maximum height and when it lands.

  1. vertex at t = −20/(2 × (−5)) = 2 s M1
  2. h = 40 − 20 = 20 m A1
  3. lands when 20t − 5t2 = 0 ⇒ t = 4 s (t = 0 is the launch) A1
GDC (calculator) methodPaper 2: graph the quadratic and use zero and maximum/minimum, or the polynomial solver. Exact answers (surds) still need the formula.
Linking questionThe discriminant reappears whenever a line meets a curve (tangents), and in complex numbers (HL), where Δ < 0 gives a conjugate pair of complex roots.
Idea for your IA / explorationModel a real parabola: a fountain or a basketball shot from a video (with free tracking software), fit y = ax2 + bx + c and compare the fitted vertex with the measurement.

Test yourself — cover the answers first

Quick check

  1. State the axis of symmetry of y = 2x2 − 8x + 1.
  2. Write x2 + 10x in completed-square form.
  3. What does Δ < 0 mean for the graph?
  4. Solve x2 = 5x.
  5. Find the roots of (x − 4)(2x + 1) = 0.

Exam-style

  1. The quadratic f(x) = x2 − 2kx + (k + 6) has no real roots. Find the range of k.
  2. A rectangle has perimeter 40 cm. Find the dimensions that give the maximum area and that area.
Show answers
  1. x = 2
  2. (x + 5)2 − 25
  3. it does not cross or touch the x-axis
  4. x = 0 or x = 5
  5. x = 4, x = −½
  6. Δ = 4k2 − 4(k + 6) < 0
    k2 − k − 6 < 0 ⇒ (k − 3)(k + 2) < 0
    −2 < k < 3
  7. sides x and 20 − x: A = x(20 − x)
    vertex at x = 10
    10 cm by 10 cm, area 100 cm2
Practise this chapter
IB Maths AA practice papers
Next chapterTransformations of graphs →
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.