Quadratic Equations

1Summary 重點摘要

A quadratic equation ax² + bx + c = 0 (a≠0) has two solutions. In S3 the main method is factorising: write it as (x−p)(x−q) = 0, then x = p or x = q. You MUST make one side equal to 0 before factorising.

2Vocabulary cards Vocabulary

3Step by step Steps

1. What a quadratic equation isAn equation in one unknown whose highest power is 2 is a quadratic equation, with standard form ax² + bx + c = 0 where a is not zero, for example x² + 5x + 6 = 0. It is called 'quadratic' because the highest power of x is 2, and 'in one variable' because only x appears. If a = 0 the x² term vanishes and the equation degrades to a linear one, no longer quadratic. Identifying a, b and c is the first step in choosing a solving method.
2. Step one: rearrange to standard formBefore solving, move every term to one side so the other side is 0, giving the standard form. From x² + 5x = −6, bringing −6 across as +6 gives x² + 5x + 6 = 0, so a = 1, b = 5, c = 6. Signs flip whenever a term crosses the equals sign: positive becomes negative and vice versa. Common errors are forgetting the sign change or omitting the = 0, which blocks factorisation and the zero-product property and stalls the whole problem.
3. Step two: factoriseFor equations of the form x² + px + q, hunt for two numbers with product q and sum p. In x² + 5x + 6 = 0 we need two numbers multiplying to 6 and adding to 5; testing shows 2 × 3 = 6 and 2 + 3 = 5, so the factorisation is (x + 2)(x + 3) = 0. If the pair is not obvious, list all factor pairs of q and check their sums. When the product is positive but the sum negative, both numbers are negative, as −2 and −5 in x² − 7x + 10.
4. Step three: apply the zero-product propertyThe zero-product property says that if two numbers multiply to 0, at least one of them is 0. Hence (x + 2)(x + 3) = 0 means x + 2 = 0 or x + 3 = 0 — either condition alone may hold. This is the key that turns a factorised quadratic into two roots; a product equal to any other number, say 12, cannot be split this way. A frequent mistake is writing x = −2 and x = −3 with 'and'; the correct word is 'or', since x takes one value at a time.
5. Step four: solve for both rootsSolve each linear equation separately: x + 2 = 0 gives x = −2 and x + 3 = 0 gives x = −3, so the roots are x = −2 or x = −3; both must be stated, since omitting either makes the answer incomplete. A quadratic generally has two roots (possibly equal; complex cases await senior forms, and junior work usually deals with two real roots). Join the roots with 'or' and keep the notation consistent, for example always writing x =.
6. Checking and the recurring errorsSubstitute each root back: for x = −2, (−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0; for x = −3, (−3)² + 5(−3) + 6 = 9 − 15 + 6 = 0; both hold, so the answer is correct. The recurring errors are: no sign change when moving terms; matching the product but ignoring the sum; writing 'and' instead of 'or'; and dropping one of the two roots. Building the habit of substitution exposes all four mistakes instantly — the final line of defence for steady marks.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• First rearrange to standard form: ax² + bx + c = 0
• Factorise: find two numbers (add = b, multiply = c)
• Zero product: A×B=0 → A=0 or B=0
• Remember to flip sign: x+2=0 → x=−2
• Quadratic usually has 2 roots, don't miss one