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SSC CGL Maths · Algebraic Identities and Simplification

Squares of a Binomial

The square of a sum or a difference, the difference of two squares, and how the sum and product of two numbers give every other square.

All Algebraic Identities and Simplification notes

Why this matters

20 questions in the bank, 16 of them EASY. Each one is a single square identity used once: expand a bracket, or turn a sum and a product into a sum of squares, a difference or a difference of squares. Because an identity holds for every value, putting a small number into the question and the options settles most of them in seconds.

Concept 1 of 3: Checking an algebra answer by putting in numbers

An identity is true for every value of its letters. So two expressions that are really equal must give the same number for any value you put in. That turns an algebra question into a small sum.

Definition

  • Pick small values that keep every term defined: x=1x = 1, x=2x = 2, or a=2,b=1a = 2, b = 1. Avoid 0 when something is divided by xx.
  • Work out the question and each option at those values. Throw out every option that gives a different number.
  • If two options survive, try a second value.
  • Limit: a mismatch proves two expressions differ, but a match at one value proves nothing. Use numbers to reject options and the identity to be sure.
  • Limit: this works when the options are expressions in the letters. If the question fixes a condition (say x+y=7x + y = 7), any test values must satisfy that condition too.

Worked example

Which of 4x4x, 8x8x, 8x28x^2 and 2 is equal to (2x+1)2−(2x−1)2(2x + 1)^2 - (2x - 1)^2?
Practice this conceptself-check · 4 quick reps

A test value that hides the difference

At x=1x = 1, xx, x2x^2 and x3x^3 are all 1, so 4x4x and 4x24x^2 give the same number. When options differ only in a power, test at x=2x = 2 as well.

Concept 2 of 3: Square of a sum, square of a difference, difference of squares

(a+b)2(a + b)^2 is the area of a square of side a+ba + b: a square of side a, a square of side b, and two rectangles a by b. Those two rectangles are the 2ab2ab, and leaving them out is the most common slip.

Definition

  • (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 and (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
  • Add the two and the 2ab2ab cancels: (a+b)2+(a−b)2=2(a2+b2)(a + b)^2 + (a - b)^2 = 2(a^2 + b^2).
  • Subtract them and the squares cancel: (a+b)2−(a−b)2=4ab(a + b)^2 - (a - b)^2 = 4ab.
  • Difference of squares: a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b). Read it both ways: to factorise, and to multiply two numbers the same distance from a round number, such as 98×102=1002−22=999698 \times 102 = 100^2 - 2^2 = 9996.
  • A long bracket is still a binomial. In (3x−2y+5)(3x−2y−5)(3x - 2y + 5)(3x - 2y - 5), let a=3x−2ya = 3x - 2y and b=5b = 5.
  • For a product that is not a square: (x+p)(x+q)=x2+(p+q)x+pq(x + p)(x + q) = x^2 + (p + q)x + pq. Expand each product, then collect like terms.

Square identities

(a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2(a+b)2+(a−b)2=2(a2+b2)(a+b)2−(a−b)2=4aba2−b2=(a−b)(a+b)(a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2 \qquad (a + b)^2 + (a - b)^2 = 2(a^2 + b^2) \qquad (a + b)^2 - (a - b)^2 = 4ab \qquad a^2 - b^2 = (a - b)(a + b)

Worked example

Simplify (2x+5)2+(2x−5)2(2x + 5)^2 + (2x - 5)^2.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 26 Sep 2025, Shift 3 · Q70Easy

Example 2 · Algebraic Identities and Simplification · Squares of a Binomial

Given, x=3x = \sqrt{3}, what is (x+1)2+(x−1)2(x + 1)^2 + (x - 1)^2 ?

Squaring term by term

(a+b)2(a + b)^2 is not a2+b2a^2 + b^2: the 2ab2ab is missing. At a=b=1a = b = 1 the first is 4 and the second is 2.

The last term of a squared difference

(x−5)2=x2−10x+25(x - 5)^2 = x^2 - 10x + 25. The last term is +25+25, never −25-25, because it is (−5)2(-5)^2. Only the middle term carries the minus sign.
Drill 10 more on square of a sum, square of a difference, difference of squares

Concept 3 of 3: Squares from the sum and product of two numbers

If you know a+ba + b and abab, you know every symmetric expression in the two numbers without finding them. The bridge is that (a+b)2(a + b)^2 and (a−b)2(a - b)^2 differ by exactly 4ab4ab.

Definition

  • a2+b2=(a+b)2−2ab=(a−b)2+2aba^2 + b^2 = (a + b)^2 - 2ab = (a - b)^2 + 2ab.
  • (a−b)2=(a+b)2−4ab(a - b)^2 = (a + b)^2 - 4ab and (a+b)2=(a−b)2+4ab(a + b)^2 = (a - b)^2 + 4ab.
  • a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b): find whichever of the two is missing first.
  • a2−ab+b2=(a+b)2−3aba^2 - ab + b^2 = (a + b)^2 - 3ab, and ab+ba=a2+b2ab\frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab}.
  • A square root has two signs. Use what the question says (natural numbers, positive, a>ba > b, a sign given) to pick one.
  • A coefficient rides along. For 2x+5y2x + 5y and xyxy, take a=2xa = 2x, b=5yb = 5y, so ab=10xyab = 10xy.
  • Fast route: with whole numbers, guess the pair. A sum of 9 and a product of 20 are 5 and 4. If the guess fits both facts, the rest is arithmetic. If no whole pair fits, use the identities.

From a sum and a product

a2+b2=(a+b)2−2ab(a−b)2=(a+b)2−4aba2−ab+b2=(a+b)2−3abab+ba=a2+b2aba^2 + b^2 = (a + b)^2 - 2ab \qquad (a - b)^2 = (a + b)^2 - 4ab \qquad a^2 - ab + b^2 = (a + b)^2 - 3ab \qquad \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab}

Worked example

The sum of two natural numbers is 17 and their product is 72. Find the positive difference of their squares.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 23 Sep 2024, Shift 3 · Q69Easy

Example 3 · Algebraic Identities and Simplification · Squares of a Binomial

If 3x+2y=103x + 2y = 10 and 2xy=72xy = 7, then find the value of 3x−2y3x - 2y (if 3x−2y>03x - 2y > 0).

Leaving out the coefficient in the product

With a=3xa = 3x and b=4yb = 4y, ab=12xyab = 12xy, not xyxy. Using xyxy in (a−b)2=(a+b)2−4ab(a - b)^2 = (a + b)^2 - 4ab gives a wrong difference.

Two square roots, one answer

(a−b)2=49(a - b)^2 = 49 gives a−b=7a - b = 7 or −7-7. The question's condition (a>ba > b, natural numbers, a positive result) picks one. Without a condition, see which sign the options allow.
Drill 8 more on squares from the sum and product of two numbers

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Square of a sum, square of a difference, difference of squares

    Square identities

    (a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2(a+b)2+(a−b)2=2(a2+b2)(a+b)2−(a−b)2=4aba2−b2=(a−b)(a+b)(a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2 \qquad (a + b)^2 + (a - b)^2 = 2(a^2 + b^2) \qquad (a + b)^2 - (a - b)^2 = 4ab \qquad a^2 - b^2 = (a - b)(a + b)
  • Squares from the sum and product of two numbers

    From a sum and a product

    a2+b2=(a+b)2−2ab(a−b)2=(a+b)2−4aba2−ab+b2=(a+b)2−3abab+ba=a2+b2aba^2 + b^2 = (a + b)^2 - 2ab \qquad (a - b)^2 = (a + b)^2 - 4ab \qquad a^2 - ab + b^2 = (a + b)^2 - 3ab \qquad \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab}

Watch out for (5)

Test yourself on Algebraic Identities and Simplification

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Algebraic Identities and Simplification chapter testDrill all 20 questions