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Quadratics – The number of roots of a quadratic equation

📘 Notes

Number of Roots of a Quadratic (Year 12)

We use the discriminant to decide how many real solutions a quadratic equation has.

1. The Discriminant

For a quadratic equation:

\[ ax^2 + bx + c = 0 \]

The discriminant is:

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

The value of the discriminant tells us how many roots the quadratic has:

Value of \( \Delta \) Number of Real Roots
\(\Delta > 0\)Two distinct real roots
\(\Delta = 0\)One repeated real root
\(\Delta < 0\)No real roots

2. Example 1 — Two Real Roots

How many real roots does

\[ 3x^2 - 5x + 2 = 0 \]

Compute the discriminant:

\[ \Delta = (-5)^2 - 4(3)(2) = 25 - 24 = 1 > 0 \]

Two distinct real roots.

3. Example 2 — One Repeated Root

How many real roots does \[ x^2 - 6x + 9 = 0 \]

\[ \Delta = (-6)^2 - 4(1)(9) = 36 - 36 = 0 \]

One repeated root (the quadratic touches the x-axis).

4. Example 3 — No Real Roots

How many real roots does \[ 2x^2 + 4x + 10 = 0 \]

\[ \Delta = 4^2 - 4(2)(10) = 16 - 80 = -64 < 0 \]

No real roots (the graph is above the x-axis).

5. Parameters and the Discriminant

For: \[ x^2 + (k - 3)x + 4 = 0 \] find the values of \(k\) for which there are no real roots.

Use: \[ \Delta < 0 \]

\[ \Delta = (k - 3)^2 - 4(1)(4) < 0 \Rightarrow (k - 3)^2 < 16 \]

\[ -4 < k - 3 < 4 \Rightarrow -1 < k < 7 \]

No real roots when \( -1 < k < 7 \).

6. Tips for Year 12

  • A quadratic always has 2, 1 or 0 real roots — never 3.
  • The discriminant always determines the number of real roots.
  • If \(a > 0\), the graph opens upwards; if \(a < 0\), it opens downwards.
  • If \(\Delta = 0\), the curve touches the x-axis at one point.
  • If \(\Delta < 0\), the graph does not touch the x-axis.

7. Try These

  1. Find the number of roots of \(5x^2 - 4x + 1 = 0\).
  2. For what values of \(p\) does \(x^2 + px + 9 = 0\) have two real roots?
  3. For what values of \(k\) does \(2x^2 + kx + 8 = 0\) have exactly one real root?

Use the discriminant in each case.