Quadratic Equations, Discriminant & Inequalities
Master solving quadratic equations by factoring and formula, the discriminant and nature of roots, quadratic inequalities, and simultaneous equations — with full worked examples.
Equations and Inequalities
Table of Contents
Equations & Inequalities: Quadratics, Discriminant & More#
In Grade 10, you solved linear equations ($ax + b = 0$). In Grade 11, the highest power becomes 2 — and that changes everything. A quadratic equation can have two, one, or no real solutions, and you need multiple methods to handle them all.
The Three Methods for Solving Quadratic Equations#
Method 1: Factorising (fastest, but doesn’t always work)#
$$x^2 + 5x + 6 = 0 \Rightarrow (x + 2)(x + 3) = 0$$Zero product rule: If $A \times B = 0$, then $A = 0$ or $B = 0$.
$x + 2 = 0$ → $x = -2$ or $x + 3 = 0$ → $x = -3$
⚠️ NEVER divide both sides by $x$. You’ll lose the solution $x = 0$. Always move everything to one side and factor.
Method 2: The Quadratic Formula (always works)#
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$For $ax^2 + bx + c = 0$, identify $a$, $b$, $c$ and substitute. The $\pm$ gives you both solutions in one formula.
Method 3: Completing the Square#
Rewrite $ax^2 + bx + c = 0$ so the left side becomes a perfect square $(x + p)^2 = q$, then take the square root of both sides.
The Discriminant ($\Delta$): Predicting the Roots#
$$\Delta = b^2 - 4ac$$The discriminant tells you how many solutions exist before you solve:
| $\Delta$ value | Nature of roots | What it means |
|---|---|---|
| $\Delta > 0$ (perfect square) | Two real, rational, unequal roots | Factorising will work |
| $\Delta > 0$ (not perfect square) | Two real, irrational, unequal roots | Use quadratic formula |
| $\Delta = 0$ | Two real, equal roots | The parabola just touches the x-axis |
| $\Delta < 0$ | Non-real (no real roots) | The parabola doesn’t cross the x-axis |
💡 The connection to functions: The roots of $ax^2 + bx + c = 0$ are the x-intercepts of the parabola $y = ax^2 + bx + c$. The discriminant tells you whether the parabola crosses the x-axis, touches it, or floats above/below it.
Quadratic Inequalities#
Solving $x^2 - 5x + 6 < 0$ means finding the range of $x$-values where the parabola is below the x-axis.
The method:
- Solve the equation $x^2 - 5x + 6 = 0$ → $x = 2$ or $x = 3$
- Sketch a quick parabola through these roots ($a > 0$ → U-shape)
- Read off where the parabola is below the x-axis: $2 < x < 3$
| Inequality | Where to look on the parabola |
|---|---|
| $f(x) < 0$ or $f(x) \leq 0$ | Below the x-axis |
| $f(x) > 0$ or $f(x) \geq 0$ | Above the x-axis |
Simultaneous Equations (One Linear + One Quadratic)#
In Grade 11, simultaneous equations involve a straight line and a parabola (or circle). The strategy:
- Solve the linear equation for $y$ (or $x$).
- Substitute into the quadratic equation.
- You get a quadratic in one variable — solve it.
- Back-substitute to find the other variable.
Deep Dive#
- Quadratic Equations, Discriminant & Inequalities — full worked examples for all methods, nature of roots, inequalities with sign diagrams, and simultaneous equations
🚨 Common Mistakes#
- Dividing by $x$: NEVER divide both sides by $x$ (or $\sin\theta$, etc.). Move everything to one side and factor. Dividing loses the $x = 0$ solution.
- Sign errors in the quadratic formula: Watch $b^2 - 4ac$ carefully. If $c = -7$, then $-4ac = -4(a)(-7) = +28a$.
- Inequality sign after multiplying by $-1$: The sign FLIPS. $-x^2 + 3 > 0$ becomes $x^2 - 3 < 0$.
- Simultaneous equations — expanding $(x+1)^2$: It’s $x^2 + 2x + 1$, NOT $x^2 + 1$. The middle term is crucial.
- Forgetting restrictions: $\frac{3}{x-2}$ is undefined when $x = 2$. State restrictions upfront.
🔗 Related Grade 11 topics:
- The Parabola — the x-intercepts of a parabola = the roots of the quadratic equation
- Surds & Exponential Equations — surd equations become quadratics after squaring
- Trig Identities & Equations — trig equations often reduce to quadratics
📌 Grade 10 foundation: Solving Equations & Inequalities
📌 Grade 12 extension: Algebra, Equations & Inequalities
⏮️ Exponents & Surds | 🏠 Back to Grade 11 | ⏭️ Number Patterns