Quadratic Inequalities

To solve a quadratic inequality, never trust sign rules recited from memory, sketch. The method is three steps and it is essentially error-proof:

The method

Solve $2x^2 - 5x - 3 \ge 0$.

1. Roots: solve the equality. $2x^2 - 5x - 3 = 0 \to (2x + 1)(x - 3) = 0 \to x = -\frac{1}{2}, 3$
2. Sketch: positive $x^2$ coefficient $\to$ upward parabola crossing at $-\frac{1}{2}$ and $3$
3. Read the region: "$\ge 0$" means on or above the axis, the outer arms. $x \le -\frac{1}{2}$ or $x \ge 3$

The factorising line and the sketch are both visible method; the final answer takes the marks only if the regions are right, which the sketch guarantees.

Writing the answer properly

• Outside regions ($> 0$ with upward parabola): two separate inequalities joined by “or”, $x \le -\frac{1}{2}$ or $x \ge 3$. The chained form "$-\frac{1}{2} \ge x \ge 3$" is meaningless and scores nothing.
• Inside region ($< 0$): one sandwich, $-\frac{1}{2} < x < 3$.
• Strict vs non-strict follows the question’s symbol: $\ge$ keeps the roots, $>$ excludes them.

A downward parabola (negative $x^2$ coefficient) flips which region is which, another reason the sketch beats memorised rules. Alternatively multiply through by $-1$ first and flip the inequality sign.

Where these inequalities ambush you

The most common 0606 appearance isn’t a standalone question, it’s the second half of a discriminant problem:

“Find the values of $k$ for which $x^2 + kx + 2k = 0$ has two distinct real roots.” $b^2 - 4ac > 0$: $k^2 - 8k > 0 \to k(k - 8) > 0 \to$ sketch in $k$ $\to$ $k < 0$ or $k > 8$

Students who can do both halves separately still lose marks by treating the $k$-inequality as an equation and stopping at $k = 0, 8$. The variable changed name; the method didn’t.

Common mistakes

• Regions read without a sketch (the wrong-way-round answer)
• Two-region answers chained into one impossible inequality
• Roots included/excluded against the question’s symbol
• Discriminant questions left as equalities
• Dividing by a negative without flipping the sign

Full topic context: Quadratic Functions notes · cubic versions: solving cubic inequalities graphically.