Discriminant

\( \Delta = b^2 - 4ac \)
Algebra GCSE
Question 9 of 20
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\( Given\; ax^2+bx+c=0\; with a=4,\;b=-12,\;c=9.\; Compute \Delta and state the nature of the roots (enter \"0\" for \Delta). \)

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
Square b then subtract 4ac

Explanation

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Statement

The discriminant is a key part of the quadratic formula and helps determine the nature of the solutions of a quadratic equation of the form \(ax^2 + bx + c = 0\). It is given by:

\[\Delta = b^2 - 4ac\]

The value of \(\Delta\) tells us how many and what type of solutions (roots) the quadratic equation has.

Why it’s true

  • The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
  • The expression inside the square root, \(b^2 - 4ac\), is the discriminant.
  • If the discriminant is positive, the square root is a real number, giving two distinct real roots.
  • If it is zero, the square root vanishes, giving one repeated real root.
  • If it is negative, the square root is imaginary, giving two complex conjugate roots.

Recipe (how to use it)

  1. Identify coefficients \(a\), \(b\), and \(c\) from the quadratic equation \(ax^2 + bx + c = 0\).
  2. Calculate \(\Delta = b^2 - 4ac\).
  3. Interpret the result:
    • \(\Delta > 0\): two real and distinct roots.
    • \(\Delta = 0\): one real repeated root.
    • \(\Delta < 0\): two complex roots.

Spotting it

The discriminant is relevant whenever you are asked about the number or type of solutions of a quadratic, without needing to solve it fully.

Common pairings

  • Quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\).
  • Completing the square (where the discriminant shows whether the square term balances).
  • Graph of a parabola (the discriminant tells how many times it cuts the x-axis).

Mini examples

  1. Given: \(x^2 - 5x + 6 = 0\). Find: discriminant. Answer: \(b^2 - 4ac = 25 - 24 = 1 > 0\), so two real roots.
  2. Given: \(x^2 + 4x + 4 = 0\). Find: discriminant. Answer: \(16 - 16 = 0\), one repeated root.
  3. Given: \(x^2 + 2x + 5 = 0\). Find: discriminant. Answer: \(4 - 20 = -16 < 0\), two complex roots.

Pitfalls

  • Forgetting to square \(b\) correctly.
  • Using the wrong signs for \(a\), \(b\), or \(c\).
  • Mixing up the discriminant with the quadratic formula itself.

Exam strategy

  • Quickly calculate \(\Delta\) before deciding how to solve the quadratic.
  • If \(\Delta\) is negative, don’t waste time looking for real roots.
  • If \(\Delta\) is a perfect square, expect rational roots.

Summary

The discriminant \(\Delta = b^2 - 4ac\) is a shortcut to determine the nature of quadratic roots. It helps decide whether solutions are real or complex, distinct or repeated, and links algebra with the graph of the parabola.

Worked examples

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  1. \( Find the discriminant of x^2 + 3x + 2 = 0 and state the nature of its roots. \)
    1. \( a=1, b=3, c=2 \)
    2. \( Δ = b^2 - 4ac = 9 - 8 = 1 \)
    3. Since Δ>0, two real distinct roots
    Answer: Two real distinct roots
  2. \( Find the discriminant of x^2 + 4x + 4 = 0 and state the nature of its roots. \)
    1. \( a=1, b=4, c=4 \)
    2. \( Δ = 16 - 16 = 0 \)
    3. One repeated real root
    Answer: One real repeated root
  3. \( Find the discriminant of x^2 + 2x + 5 = 0 and describe its roots. \)
    1. \( a=1, b=2, c=5 \)
    2. \( Δ = 4 - 20 = -16 \)
    3. Δ<0, two complex roots
    Answer: Two complex roots
  4. \( For the quadratic 2x^2 - 3x + 1 = 0, calculate the discriminant. \)
    1. \( a=2, b=-3, c=1 \)
    2. \( Δ = (-3)^2 - 4×2×1 = 9 - 8 = 1 \)
    3. Δ>0, two real distinct roots
    Answer: 1
  5. \( Find the discriminant of 3x^2 + 6x + 3 = 0 and interpret the result. \)
    1. \( a=3, b=6, c=3 \)
    2. \( Δ = 36 - 36 = 0 \)
    3. One repeated real root
    Answer: 0
  6. \( Determine the discriminant of x^2 - 6x + 13 = 0. \)
    1. \( a=1, b=-6, c=13 \)
    2. \( Δ = 36 - 52 = -16 \)
    3. Δ<0, two complex roots
    Answer: -16
  7. \( Find the discriminant of 4x^2 + 4x + 1 = 0. \)
    1. \( a=4, b=4, c=1 \)
    2. \( Δ = 16 - 16 = 0 \)
    3. One repeated real root
    Answer: 0
  8. \( For the quadratic 5x^2 - 20x + 20 = 0, calculate the discriminant. \)
    1. \( a=5, b=-20, c=20 \)
    2. \( Δ = 400 - 400 = 0 \)
    3. One repeated real root
    Answer: 0
  9. \( Find the discriminant of 2x^2 + 5x + 10 = 0. \)
    1. \( a=2, b=5, c=10 \)
    2. \( Δ = 25 - 80 = -55 \)
    3. Δ<0, two complex roots
    Answer: -55
  10. \( Determine the discriminant of x^2 - 2x - 15 = 0. \)
    1. \( a=1, b=-2, c=-15 \)
    2. \( Δ = 4 - (-60) = 64 \)
    3. Δ>0, two real distinct roots
    Answer: 64