Quadratics – Solving quadratic inequalities

1. Solve the quadratic equation to find the roots.
2. Sketch a quick sign diagram or curve shape (U or ∩).
3. Shade the regions that satisfy the inequality.
4. Use open or closed circles for strict inequalities (<, >) or inclusive (≤, ≥).

Solve: \[ x^2 - 5x + 6 > 0 \]

Step 1: Factorise:

\[ (x - 2)(x - 3) > 0 \]

Step 2: Roots are \(x = 2, 3\). Sketch or sign diagram:

Positive outside the roots (U-shape upwards).

Solve: \[ x^2 - 4x + 3 \le 0 \]

Step 1: Factorise:

\[ (x - 1)(x - 3) \le 0 \]

Step 2: Roots are \(x = 1, 3\). A ≤ sign means include the roots.

Solve: \[ -2x^2 + 3x + 5 < 0 \]

Step 1: Solve the equation:

\[ -2x^2 + 3x + 5 = 0 \Rightarrow x = -1 \text{ or } x = \frac{5}{2} \]

Step 2: Sketch an upside-down ∩ shape (because coefficient is negative).

We want where expression is negative (below x-axis).

Solve: \[ x^2 - 6x + 10 > 0 \]

Step 1: Discriminant:

\[ b^2 - 4ac = (-6)^2 - 4(1)(10) = 36 - 40 = -4 < 0 \]

Since the quadratic has no real roots and \(a = 1 > 0\), the graph is always above the x-axis.

• Always find roots by factorising or quadratic formula.
• Draw a quick sign diagram to avoid mistakes.
• For ≤ or ≥ include the roots (closed circles).
• For < or > exclude the roots (open circles).
• If no real roots, look at whether the curve is above or below the x-axis.

1. \(x^2 - 7x + 10 \ge 0\)
2. \(3x^2 - 2x - 8 < 0\)
3. \(4x^2 + 12x + 9 > 0\)

Give solutions using inequality notation.