27. Quadratic equations and roots
Free study material · concepts, shortcuts & solved questions
For ax²+bx+c=0, sum of roots=−b/a and product=c/a. Discriminant D=b²−4ac: D>0 gives distinct real roots, D=0 equal roots, D<0 non-real roots. Factorisation is fastest when factors are visible.
Method before formula Check the sign of b in the sum formula. If roots are α and β, equations can be reconstructed as x²−(α+β)x+αβ=0 for a monic quadratic. |
Formula and decision table
Equation | Roots/relations |
x²−5x+6 | Roots 2,3; sum 5, product 6 |
x²−9 | Roots ±3 |
D=0 | Repeated real root |
Worked understanding
A reliable solution has five visible moves: identify the data, choose the base or unit, write the rule, substitute carefully, and check the size of the answer. The examples in this chapter are deliberately small so that the method remains visible.
Calculation discipline Before pressing ahead, state the denominator or unit in words. For example: profit on CP, discount on MP, average over number of items, speed in metres per second, and probability over the fixed sample space. |
Solved examples from this chapter
Question focus | Correct move | Answer |
Roots of x²−5x+6=0: | Factor as (x−2)(x−3). | B. 2,3 |
For x²+4x+4, roots are: | It is (x+2)². | B. Equal |
Product of roots of ax²+bx+c: | Product=c/a. | C. c/a |
What an examiner is testing
The options around Quadratic equations and roots usually represent a wrong base, a missed conversion, a sign error or a shortcut used outside its condition. Say the base and unit before calculating, estimate the result, and use substitution or a reverse operation as the final check.
Step-by-step answer construction
Example 1: Roots of x²−5x+6=0: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 2,3, because Factor as (x−2)(x−3).
Example 2: For x²+4x+4, roots are: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is Equal, because It is (x+2)².
Example 3: Product of roots of ax²+bx+c: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is c/a, because Product=c/a.
Chapter practice
1. Roots of x²−5x+6=0:
(A) 1,6 (B) 2,3 (C) −2,−3 (D) 5,6
Answer: B. 2,3 | Explanation: Factor as (x−2)(x−3).
2. For x²+4x+4, roots are:
(A) Distinct (B) Equal (C) Non-real (D) No roots
Answer: B. Equal | Explanation: It is (x+2)².
3. Product of roots of ax²+bx+c:
(A) b/a (B) −b/a (C) c/a (D) −c/a
Answer: C. c/a | Explanation: Product=c/a.
SSC CGL speed and trap clinic
For Quadratic equations and roots, speed comes after classification. Before calculating, say aloud what the number means, what the denominator is, and what unit the answer must carry. Then use the nearest-option check to catch a sign, base or conversion error.
Checkpoint | What to verify | Typical SSC mistake |
Base | Which quantity is the base in Quadratic equations and roots? | Using selling price instead of cost price, or part instead of total |
Unit | Are time, distance, area and rate in compatible units? | Mixing hours with minutes or km/h with m/s |
Direction | Should the answer increase, decrease or stay bounded? | Accepting an impossible negative length or probability above 1 |
Estimate | What range should the answer lie in before exact work? | Trusting a long calculation that is far from the options |
Reverse check | Can the answer be substituted back into the condition? | Stopping at an algebraic value without verification |
Mini decision drill
1. Write the first operation you would perform in a Quadratic equations and roots question and why.
2. State the most dangerous denominator or unit in this chapter.
3. Give one condition under which a shortcut would be invalid.
4. Create a small numerical example and verify it by a second method.
Revision evidence Do not tick this chapter because you recognised the formula. Tick it only after you solve one direct question, one altered-condition question and one mixed-paper question correctly. |