Algebra basics for RRB Group D

Algebra in RRB Group D is Class 9–10 level and short. You will not be asked to prove anything. You will be asked to use an identity to find a value, or to solve a simple equation, often dressed up as a word problem.

That makes it one of the easiest topics to secure. The whole toolkit fits on half a page, and every answer can be checked in a few seconds with a small number. This guide gives you that toolkit, shows each type worked out, and ends with a practice set.

The identities to learn

Learn these until you can write them without thinking. Almost every algebra question uses one of them.

The forms that questions actually use

Most questions give you two pieces of information and ask for a third. Rearranging the identities gives the shortcuts:

• a² + b² = (a + b)² − 2ab
• a² + b² = (a − b)² + 2ab
• (a + b)² − (a − b)² = 4ab
• x² + 1/x² = (x + 1/x)² − 2
• x² + 1/x² = (x − 1/x)² + 2
• x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x)

Solving a linear equation, step by step

A linear equation has the unknown only to the power 1. Use the same four steps every time:

1. Clear brackets. Multiply out anything like 5(x − 2).
2. Clear fractions. Multiply every term by the LCM of the denominators.
3. Collect terms. Unknowns on one side, plain numbers on the other. A term changes sign when it crosses the = sign.
4. Divide by the number in front of x, then check by putting the answer back.

For two equations with two unknowns, add or subtract the equations so that one unknown disappears. This is called elimination, and it is quicker than substitution for exam questions.

Worked examples

Example 1: If x + 1/x = 4, find x² + 1/x² and x⁴ + 1/x⁴.

• x² + 1/x² = 4² − 2 = 16 − 2 = 14.
• Use the same idea again, one level up: x⁴ + 1/x⁴ = 14² − 2 = 196 − 2 = 194.

Example 2: If a + b = 10 and ab = 21, find a² + b² and a − b (where a > b).

• a² + b² = 10² − 2 × 21 = 100 − 42 = 58.
• (a − b)² = (a + b)² − 4ab = 100 − 84 = 16, so a − b = 4.
• Check with numbers: a and b are 7 and 3. Then 49 + 9 = 58 and 7 − 3 = 4.

Example 3: Find 53 × 47 without long multiplication.

• 53 × 47 = (50 + 3)(50 − 3) = 50² − 3² = 2,500 − 9 = 2,491.

Example 4: Solve 5(x − 2) = 3x + 6.

• Clear brackets: 5x − 10 = 3x + 6.
• Collect: 5x − 3x = 6 + 10, so 2x = 16 and x = 8.
• Check: 5 × 6 = 30 and 3 × 8 + 6 = 30.

Example 5: Solve x/3 + x/4 = 14.

• The LCM of 3 and 4 is 12. Multiply every term by 12: 4x + 3x = 168.
• 7x = 168, so x = 24.
• Check: 24/3 + 24/4 = 8 + 6 = 14.

Example 6: Three pens and two notebooks cost ₹74. One pen and one notebook together cost ₹30. Find the price of each.

• Let a pen cost p and a notebook cost n. Then 3p + 2n = 74 and p + n = 30.
• Double the second equation: 2p + 2n = 60.
• Subtract it from the first: p = 74 − 60 = ₹14. So n = 30 − 14 = ₹16.
• Check: 3 × 14 + 2 × 16 = 42 + 32 = 74.

The number-check trick

When a question gives a + b and ab, try to spot two whole numbers that fit, as in Example 2. Then work out the answer directly.

When a question asks you to simplify an expression and the options are expressions too, put a small value such as x = 1 or x = 2 into the question and into each option. Only the correct option gives the same number. Avoid x = 0 and x = 1 if they make several options equal; try x = 2 instead.

Common mistakes

• Writing (a + b)² as a² + b², missing the middle term 2ab.
• Forgetting to subtract 2 in x + 1/x questions, or subtracting it when the question uses x − 1/x (there you add 2).
• Sign errors when a term crosses the = sign.
• Multiplying only some terms by the LCM when clearing fractions. Every term, on both sides, must be multiplied.
• Stopping at x and forgetting the question asked for 2x, or for the other unknown.

Practice set

1. If x + 1/x = 5, find x² + 1/x².
2. If a + b = 8 and ab = 15, find a² + b².
3. Solve 3x − 7 = 14.
4. Solve x + y = 15 and x − y = 3.
5. If a − b = 4 and ab = 12, find a² + b².
6. If x − 1/x = 3, find x² + 1/x².
7. Find 102 × 98 using an identity.
8. If x + 1/x = 3, find x³ + 1/x³.

Answers:

1. 23. 5² − 2 = 25 − 2.
2. 34. 8² − 2 × 15 = 64 − 30. The numbers are 5 and 3.
3. x = 7. 3x = 21.
4. x = 9, y = 6. Adding gives 2x = 18.
5. 40. (a − b)² + 2ab = 16 + 24. The numbers are 6 and 2.
6. 11. With a minus sign, add 2: 3² + 2 = 9 + 2.
7. 9,996. (100 + 2)(100 − 2) = 10,000 − 4.
8. 18. 3³ − 3 × 3 = 27 − 9.

What to do next

• Write the seven identities and the six "useful forms" from memory once a day for a week.
• Solve ten x + 1/x and a + b, ab questions, checking each with real numbers where you can.
• Practise five word problems that turn into two equations, like Example 6.
• Revise squares and square roots, which make identity questions faster, then move on to geometry.