Summary and Quick Revision

Chapter at a Glance

Here is the whole of Quadratic Equations on one page — perfect for the night before your exam.

Basics

• Standard form: $ax^2 + bx + c = 0$, $a \neq 0$.
• A root is a value of $x$ satisfying the equation; a quadratic has at most two roots.
• The roots are the zeroes of $ax^2 + bx + c$.

Remember: Always rearrange to standard form before identifying $a$, $b$, $c$ or applying any method.

The Three Solving Methods

1. Factorisation

Split the middle term: find two numbers with product $a \times c$ and sum $b$; factor and use the zero-product rule.

2. Completing the square

Make the $x^2$ coefficient 1, then add $\left(\dfrac{b}{2}\right)^2$ to form $(x + p)^2 = q$; take square roots ($\pm$).

3. Quadratic formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Key Point: Factorisation is fastest when it works; the formula always works. Completing the square underlies the formula.

The Discriminant — Nature of Roots

The discriminant is $D = b^2 - 4ac$.

Discriminant	Nature of roots
$D > 0$	Two distinct real roots
$D = 0$	Two equal real roots, $x = -\dfrac{b}{2a}$
$D < 0$	No real roots

Key Point: For 'find $k$' problems: equal roots ⇒ $D = 0$; distinct real ⇒ $D > 0$; real (allowing equal) ⇒ $D \geq 0$; no real roots ⇒ $D < 0$.

[Board Important] 'Nature of roots' just needs $D$ — you do not have to solve the equation.

Word-Problem Cues

• Consecutive integers: $x, x+1$; consecutive even/odd: $x, x+2$.
• Area: rectangle $l \times b$; triangle $\dfrac{1}{2} \times \text{base} \times \text{height}$.
• Pythagoras: $a^2 + b^2 = c^2$ for right-triangle problems.
• Speed–time: Time = Distance ÷ Speed; set up the time difference.
• Boat/stream: upstream $= x - y$, downstream $= x + y$.
• Reciprocal: multiply by $x$ to clear the denominator.
• Two taps/pipes: combined rate $= \dfrac{1}{x} + \dfrac{1}{x \pm a}$.

Key Point: After solving, always reject inadmissible roots (negative lengths, ages, speeds) and answer in words with units.

Last-Minute Tips and Common Traps

• Rearrange to standard form first; watch the sign of $b$ when substituting into the formula.
• When completing the square with $a \neq 1$, divide by $a$ before adding $(b/2)^2$.
• Don't forget the $\pm$ when taking the square root.
• A negative discriminant in a word problem means the situation is impossible — state that clearly.
• For equal roots, the repeated root is $x = -\dfrac{b}{2a}$.
• Always reject roots that don't fit the real-world context.

Final Word: Quadratic Equations is a high-scoring, versatile chapter. Master the three methods, the discriminant rule, and the word-problem setups, and you secure most of the marks. All the best!