Class 10 Quadratics: Choose a Method From the Equation

A short decision route

1. 1

   Standard form

   Expand and collect terms on one side; identify a, b and c including their signs.

2. 2

   Inspect the factors

   If integer factors are easy to find, factorisation often gives a short solution.

3. 3

   Check the discriminant

   D = b² − 4ac tells you whether there are two, one repeated, or no real roots.

4. 4

   Use the formula

   When factors are awkward, x = (−b ± √D)/(2a) gives the real roots if D ≥ 0.

Three decisions, not another practice bank

Compare x² − 7x + 10 = 0 (obvious integer factors), x² − 4x + 1 = 0 (formula gives 2 ± √3), and a rectangle with width x and length x+3, area 40. The last model gives x²+3x−40 = 0, so (x+8)(x−5)=0; retain x=5 cm and reject x=−8 cm because a width must be positive. All three are quadratic equations, but the efficient method and final interpretation differ.

Before calculating, name your choice and why. After calculating, say whether both algebraic roots fit the original job. This guide compares decisions; the linked practice page supplies an independent attempt.

An equation that rewards factorising

For x² − 7x + 10 = 0, find two numbers with product 10 and sum −7: −5 and −2. Thus (x − 5)(x − 2) = 0, giving x = 5 or x = 2. The zero-product rule applies because the product equals zero. It does not apply to (x − 5)(x − 2) = 6.

An equation where the formula is clearer

For x² − 4x + 1 = 0, D = 16 − 4 = 12. Hence x = (4 ± √12)/2 = 2 ± √3. These are two distinct real roots even though they are not integers. Keep both signs until you interpret the context.

A negative discriminant means no real roots, not that an arithmetic mistake must have happened. A zero discriminant means the two formula expressions coincide. Class 10 real-root problems do not require you to invent a real value for the square root of a negative number.

Return to the story

If an equation for a length gives 8 and −3, only 8 can represent that positive length. Show both algebraic roots and state why one is rejected. For an age, time or count, check the original conditions rather than rejecting every negative value by habit.

Finish by substituting into the original equation, especially if you expanded brackets or multiplied through denominators. A method that is quick to perform is not necessarily quick to verify; reserve one line for that check.

Choose what to study next

Sources and scope

This is a focused learning resource, not a complete chapter or a prediction of board questions. Use your current syllabus and the paper-specific marking scheme for assessment requirements.