What you'll learn
- How to solve linear, quadratic and simultaneous equations.
- How to work with inequalities and represent solutions on a number line.
- The nature of roots using the discriminant.
Key ideas
An equation states that two expressions are equal. An inequality states that one is greater or less than the other. When you multiply or divide an inequality by a negative number, the inequality sign flips.
The quadratic formula
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
The expression under the root, $b^2 - 4ac$, is the discriminant and tells you how many real roots there are.
| Discriminant | Nature of roots |
|---|---|
| $b^2 - 4ac > 0$ | Two distinct real roots |
| $b^2 - 4ac = 0$ | One repeated real root |
| $b^2 - 4ac < 0$ | No real roots |
Worked example
Solve $x^2 - 5x + 6 = 0$.
Factorise: $(x - 2)(x - 3) = 0$, so $x = 2$ or $x = 3$.
Common mistakes
- Forgetting to flip the inequality sign when multiplying by a negative.
- Cancelling terms incorrectly across an equation.
Quick summary
- Quadratic formula solves any quadratic.
- The discriminant tells you the nature of the roots.
- Always check inequality sign direction.
Practice
- Solve $x^2 + 2x - 8 = 0$.
- Solve $2x + 3 > 11$.
Answers: 1) $x = 2$ or $x = -4$. 2) $x > 4$.
Official EduRank Team notes. Always cross-check with your textbook and teacher.