Linear and Quadratic equations, Logarithms, Progressions

Algebra uses letters to stand for unknown numbers, so that relationships can be written as equations and solved. This topic covers the standard equation types and sequences that appear in aptitude tests.

Linear equations and simultaneous equations

• A linear equation in one variable, such as ax + b = c, is solved by doing the same operation on both sides until x is alone.
• Two equations in two variables can be solved by elimination (add or subtract the equations to remove one variable) or by substitution.
• For word problems, let the unknown be x, write the conditions as equations, and then solve.

Quadratic equations

• A quadratic equation has the form ax^2 + bx + c = 0, with a not equal to 0.
• Solve it by factorisation or by the formula x = (-b ± √(b^2 - 4ac)) / 2a.
• The discriminant D = b^2 - 4ac tells the nature of the roots: two distinct real roots if D > 0, equal roots if D = 0, and no real roots if D < 0.
• The sum of the roots is -b/a and the product of the roots is c/a.

Logarithms

• log_b(x) = y means b^y = x.
• log(mn) = log m + log n, and log(m/n) = log m - log n.
• log(m^k) = k log m, and log_b(b) = 1.
• The log of 1 to any base is 0.

Progressions

• Arithmetic progression (AP): a constant difference d. The nth term is a + (n - 1)d, and the sum of n terms is n/2 x (2a + (n - 1)d).
• Geometric progression (GP): a constant ratio r. The nth term is ar^(n - 1), and the sum of n terms is a(r^n - 1) / (r - 1) for r not equal to 1.

How to solve questions

1. Identify the type of equation or sequence.
2. Write down the relevant formula or form the equation.
3. Solve step by step, keeping both sides balanced.
4. Substitute your answer back to check it.