Algebraic manipulation

Explanation

Read the structure before calculating

An expression is built from terms separated by addition or subtraction signs. In 7x^2 - 3xy + 5, the terms are 7x^2, -3xy and 5; their numerical coefficients are 7, -3 and 5. Keeping the sign with each term prevents many errors.

Like terms have exactly the same variable part, including the same powers. For example, 4x^2y and -9x^2y are like terms, but x^2y, xy^2 and x^2 are unlike. Only coefficients of like terms may be added or subtracted.

Explanation

Expansion uses the distributive law

Expanding removes brackets without changing the value. Every term outside a bracket multiplies every term inside it: k(a + b - c) = ka + kb - kc. The multiplier includes its sign, so a negative multiplier reverses the signs inside.

For two or more brackets, form every possible product by choosing one term from each bracket, then collect like terms. Writing products in an organised order makes omitted terms and sign errors easier to spot.

Explanation

Factorisation reverses expansion

To factorise is to rewrite a sum or difference as a product. Start by checking every term for a greatest common numerical factor and for variables common to all terms. Extracting less than the greatest common factor may leave the answer only partly factorised.

A factorisation is reliable when expanding it reproduces the original expression exactly. This reverse check tests coefficients, powers and signs at once and is especially valuable after several factorisation stages.

Explanation

Recognise two important identities

A difference of squares follows A^2 - B^2 = (A - B)(A + B). It applies only to a subtraction of two square terms. For example, 16x^2 - 25 is (4x)^2 - 5^2, so its factors use 4x and 5.

Perfect-square trinomials follow A^2 + 2AB + B^2 = (A + B)^2 and A^2 - 2AB + B^2 = (A - B)^2. Check both square end terms and the signed middle term before using the pattern.

Explanation

Factorise quadratic trinomials systematically

For x^2 + bx + c, seek two integers whose product is c and whose sum is b. They become the constants in (x + r)(x + s). The signs of r and s must satisfy both conditions.

For ax^2 + bx + c with a not equal to 1, one dependable method is to split bx into two terms whose coefficients multiply to ac and add to b. Then factorise the resulting four terms by grouping.

Explanation

Cubic expressions may need two stages

An expression of the form ax^3 + bx^2 + cx has x as a common factor. Extract it first, producing x(ax^2 + bx + c), and then inspect the quadratic factor for further factorisation.

Stopping after the common factor is incomplete when the quadratic can still be factorised. Conversely, do not invent real-number or fractional factors when the intended integer factorisation has already ended.

Explanation

Completing the square reveals a shifted square

For x^2 + bx + c, half the coefficient of x and square it: x^2 + bx = (x + b/2)^2 - (b/2)^2. Then include c. The subtraction compensates for the extra square that was introduced.

When the coefficient of x^2 is not 1, factor it from the x^2 and x terms before completing the square inside the bracket. Keep any constant outside the bracket separate until the final simplification.

Revision checklist and error traps

First identify the requested form: simplify means collect like terms, expand means remove brackets, factorise means produce a product, and complete the square means rewrite a quadratic using a squared binomial plus or minus a constant.

For expansion, multiply every required pair and carry signs carefully. For factorisation, check a greatest common factor before patterns or grouping, and continue until no intended factor can be split further.

Verify every final rewrite by expanding it. Do not combine unlike powers, lose a negative sign, stop at a partial factorisation, or treat algebraic fractions and equation-solving procedures as part of this section 2.2 pack.