Key facts

  • A quadratic equation must first be put in standard form ax^2 + bx + c = 0 with a not equal to zero.
  • The quadratic discriminant D = b^2 - 4ac tells whether roots are distinct real, equal real, or not real in the usual exam setting.
  • For real logarithms, the base must be positive and not 1, and the argument must be positive.
  • Base conversion gives log_a b = log_c b divided by log_c a and supports shortcuts such as log_a b times log_b a = 1.

Key Points at a Glance

  1. 1

    A linear equation in one variable is solved by isolating the variable while preserving equality on both sides.

  2. 2

    Transposition is a shortcut for balanced operations; signs and inverse operations must be changed correctly.

  3. 3

    Fractional linear equations should usually be cleared by multiplying every term by the least common multiple of denominators.

  4. 4

    In word problems, define the variable first, translate each condition into an equation, solve, and then check whether the answer fits the story.

  5. 5

    A solution of a two-variable linear equation is an ordered pair, so (x, y) and (y, x) are not automatically the same.

  6. 6

    Two simultaneous linear equations can be solved by elimination when coefficients can be matched, or by substitution when one variable is easy to express.

  7. 7

    A quadratic equation must first be put in standard form ax^2 + bx + c = 0 with a not equal to zero.

  8. 8

    Factorisation uses the zero-product rule: if a product of two factors is zero, at least one factor must be zero.

  9. 9

    The quadratic discriminant D = b^2 - 4ac tells whether roots are distinct real, equal real, or not real in the usual exam setting.

  10. 10

    The quadratic formula gives roots for any quadratic equation in standard form, even when factorisation is not obvious.

  11. 11

    A logarithm rewrites an exponent relation: a^x = N is equivalent to log_a N = x.

  12. 12

    For real logarithms, the base must be positive and not 1, and the argument must be positive.

  13. 13

    Logarithm laws combine products, quotients and powers; they do not turn sums or differences inside the argument into log addition or subtraction.

  14. 14

    Base conversion gives log_a b = log_c b divided by log_c a and supports shortcuts such as log_a b times log_b a = 1.

How should LDC algebra be approached in the exam?

LDC algebra should be approached as a form-recognition and procedure-control topic: identify the equation type, apply the shortest lawful operation, and verify the answer by substitution. According to the Rajasthan Staff Selection Board's LDC Recruitment 2018 Syllabus, Paper I carries 100 maximum marks. This topic sits in the named Mathematics syllabus area for LDC level preparation: linear equations, simultaneous linear equations, quadratic equations and logarithms. Since the official paper signal for exact question subtype is weak, preparation should be procedure-centred. The goal is not to prove abstract algebra from first principles; the goal is to recognise the form, choose the fastest lawful operation, keep signs under control and verify the answer before marking the option.

An equation states that two expressions are equal. The expression may contain one variable, such as x, or two variables, such as x and y. Solving means finding the value, or values, that make the equality true. The equality sign is not decoration. Whatever operation is done to one side must be balanced on the other side, unless the same result is reached through a compact transposition rule. For example, x + 7 = 19 gives x = 12 because subtracting 7 from both sides keeps the equation balanced. In exam practice, students often say that +7 has moved to the other side and become -7. That shorthand is acceptable only if the balancing idea is understood.

The standard checking habit is substitution. After solving, place the answer back into the original equation, not into a simplified line that may already contain the mistake. If 3x - 5 = 16 gives x = 7, check the original: 3 x 7 - 5 = 21 - 5 = 16. For one-variable equations this check usually takes less time than reworking the whole solution, and it catches sign errors, denominator errors and wrong transposition.

Algebra questions in this syllabus often combine arithmetic with language. Words such as sum, difference, product, quotient, twice, thrice, exceeds and less than must be translated carefully. The phrase '5 less than a number' is x - 5, but '5 is less than a number by 12' points to x - 5 = 12. 'A number is 4 more than another number' can become x = y + 4. The first line of working should be a clean variable definition, such as 'Let the number be x' or 'Let the two numbers be x and y'. Without this line, word problems become a guessing exercise.

The main forms to recognise are simple linear equations, fractional linear equations, a pair of simultaneous linear equations, quadratic equations and logarithmic equations or simplifications. Linear means the variable has power 1 after simplification. Quadratic means the highest power is 2. A logarithm is another way to write an exponent relation. These forms should not be mixed casually. For instance, x^2 + 5x + 6 = 0 is not solved by linear transposition, and log_a m + log_a n is not equal to log_a(m + n). Form recognition is the first speed tool.

For LDC-style numeracy, write steps compactly but not invisibly. One clean line for clearing fractions, one line for collecting terms, and one line for the answer is usually enough. Avoid mental jumps when negatives and fractions appear together. Also avoid expanding expressions unless expansion helps reach standard form. A common loss of marks comes from creating bigger numbers than the question requires. In equations, the shortest correct path is normally the one that preserves structure: clear denominators, collect like terms, factor where possible, and substitute to check.