Grade 11 Maths · Algebra

Calculate the discriminant

Work out b² − 4ac for a quadratic, taking care with signs (for example for x² − 3x + 2 = 0 it is 9 − 8 = 1).

past paper questions use it: 14 past paper questions use it
videos: 3 videos

Learn it: calculate the discriminant

Pick one and watch it, then try the questions below. Videos load only when you tap play.

How to find the discriminant and label the solutions of a quadraticBrian McLogan

How To Determine The Discriminant of a Quadratic EquationThe Organic Chemistry Tutor

Discuss the nature of Roots (using the Discriminant) - Grade 11 MathematicsMatheNatic

Now try it on a real exam question

No sign-up. Takes 3 minutes.

Question 6.4 · November 2024 · Paper 1 · National (DBE) · 5 marks

In the diagram below, the graphs of

f (x) = - x^{2} + 4 x + 5

and

g

, a straight line, are drawn.

C (3; 8)

is a point of intersection of

f

and

g

E H

is drawn parallel to the y-axis, with

E

a point on

f

and

H

a point on

g

6.4 Given:

k (x) = f (x + m) = - x^{2} - 2 m x - m^{2} + 4 x + 4 m + 5

. Determine the value of

m

such that

g

is a tangent to

k

.(5)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: calculate the discriminant

Work each one out first, then open the memo and check every tick.

3. Question 1.5.1 · June 2024 · Paper 1 · Provincial · 4 marks
1.5 For which values of $p$ will $- 2 x^{2} + 4 x - 3 = - p$ :
1.5.1 have two unequal roots?(4)
Show the memo (3 ticks)Memo
1. ✓
  Setup standard form and Delta $2 x^{2} - 4 x + (3 - p) = 0$
  $Δ = b^{2} - 4 a c$
  $Δ = (- 4)^{2} - 4 (2) (3 - p)$
  1 mark · Calculate the discriminant
2. ✓
  Simplify Delta $Δ = 16 - 24 + 8 p$
  $Δ = - 8 + 8 p$
  1 mark · Calculate the discriminant
3. ✓
  Apply condition for unequal rootsFor unequal roots, $Δ > 0$
  $- 8 + 8 p > 0$
  $8 p > 8$
  $p > 1$
  2 marks · Use the discriminant to find k
4. Question 1.2 · March 2022 · Provincial · 4 marks
1.2 Given: $3 x^{2} + k x - 3 x - k = 0$ . For which values of $k$ will the equation have real roots?(4)
Show the memo (5 ticks)Memo
1. ✓
  Group terms to identify coefficients $3 x^{2} + x (k - 3) - k = 0$
  0 marks · Calculate the discriminant
2. ✓
  State condition for real roots $Δ \geq 0$
  $b^{2} - 4 a c \geq 0$
  1 mark · Use the discriminant to find k
3. ✓
  Substitute coefficients $(k - 3)^{2} - 4 (3) (- k) \geq 0$
  1 mark · Calculate the discriminant
4. ✓
  Simplify inequality $k^{2} - 6 k + 9 + 12 k \geq 0$
  $k^{2} + 6 k + 9 \geq 0$
  $(k + 3)^{2} \geq 0$
  1 mark · Calculate the discriminant
5. ✓
  Solve for kSince a square is always non-negative, this is true for all real $k$ (or based on memo interpretation, check if specific range was required, but $(k + 3)^{2} \geq 0$ is always true).
  Memo Interpretation: $k \in R$ .
  1 mark · Use the discriminant to find k
5. Question 1.3 · March 2020 · Provincial · 4 marks
1.3 Discuss the nature of the roots of the equation $2 (x - 3)^{2} + 2 = 0$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Standard form $2 (x^{2} - 6 x + 9) + 2 = 0$
  $2 x^{2} - 12 x + 18 + 2 = 0$
  $2 x^{2} - 12 x + 20 = 0$
  1 mark · Write a quadratic in standard form
2. ✓
  Substitution into the discriminant $Δ = b^{2} - 4 a c = (- 12)^{2} - 4 (2) (20)$
  1 mark · Calculate the discriminant
3. ✓
  Value of the discriminant $Δ = 144 - 160 = - 16$
  1 mark · Calculate the discriminant
4. ✓
  ConclusionSince $Δ < 0$ , roots are non-real (imaginary).
  1 mark · Determine the nature of roots
6. Question 1.3.2 · March 2018 · Provincial · 4 marks
1.3 Given: $f (x) = 9 x^{2} - 15 x + 4$
1.3.2 Show that $f (x) = - 3$ has no real roots.(4)
Show the memo (3 ticks)Memo
1. ✓
  Set up equation $9 x^{2} - 15 x + 4 = - 3$
  $9 x^{2} - 15 x + 7 = 0$
  1 mark · Write a quadratic in standard form
2. ✓
  Calculate Discriminant $Δ = b^{2} - 4 a c$
  $= (- 15)^{2} - 4 (9) (7)$
  $= 225 - 252$
  $= - 27$
  2 marks · Calculate the discriminant
3. ✓
  ConclusionSince $Δ < 0$ , the equation has no real roots.
  1 mark · Determine the nature of roots

Free on Android

Know exactly which skill is costing you marks.

Mark past papers tick by tick in the NavyBlue app. It remembers every mark you lose, shows you the skills to fix first, and picks your next questions.

The NavyBlue app showing a learner's weak skills