Quadratic Equation

1Summary 重點摘要

Solve quadratic equations, use factorisation method and formula method.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Write in standard formThe first step in solving a quadratic equation is to rearrange it into the standard form ax² + bx + c = 0, where a is not zero. Move every term to the left side so the right side becomes 0, then arrange the terms in descending powers of x and collect like terms. This matters because only after identifying the coefficients a, b and c can we factorise correctly or substitute into the formula. A common mistake is forgetting to flip the sign when moving a term across the equals sign.
2. Step 2: Try factorisationNext, try to factorise the left side into a product of two linear factors. A common technique is the cross-method: when a = 1, look for two numbers whose product is c and whose sum is b. Factorisation is preferred whenever possible because it is faster than the formula and less prone to arithmetic errors. A common mistake is skipping the check: expand the factors once to confirm the middle term matches the original expression.
3. Step 3: Zero product propertyApply the zero product property: if the product of two expressions equals 0, then at least one of them must be 0. That is, when A × B = 0, we must have A = 0 or B = 0. This property is the key that turns a product form into two linear equations. It only works when the right side is already 0. A common mistake is taking square roots of both sides or cancelling a factor, which loses a root.
4. Step 4: Solve each linear equationSet each factor equal to 0 and solve the two linear equations separately to obtain two roots. For example, from (x + 2)(x + 3) = 0 we solve x + 2 = 0 and x + 3 = 0, giving x = -2 or x = -3. A quadratic equation generally has two real roots, so both equations must be solved. A common mistake is forgetting to flip the sign when moving a constant to the other side.
5. Step 5: Use the formula when factorisation failsIf factorisation does not work, for example when the coefficients are large or the roots are irrational, use the quadratic formula x = (-b ± √(b² - 4ac)) / 2a, which applies to every quadratic equation. Before substituting, confirm the equation is in standard form and read the signs of a, b and c carefully. The nature of the roots is decided by the discriminant b² - 4ac. A common mistake is forgetting brackets around negative values of b or c.
6. Step 6: Check the answersFinally substitute both roots into the original equation to confirm the left side really equals 0. Checking reveals factorisation errors or sign slips immediately and costs very little time in an exam. Substitute into the original equation rather than a factorised intermediate form, otherwise the earliest mistake stays hidden. If a root does not give 0, recheck step by step from the standard form.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Quadratic equation highest degree 2
• First rearrange to ax² + bx + c = 0
• If can factorise use factorisation method
• Zero product: A × B = 0 → A=0 or B=0
• Can't factorise then use quadratic formula