What Is the (a + b)² Formula?
The a plus b whole square formula expands the square of a binomial $(a + b)$ into a trinomial. For any real (or even complex) numbers $a$ and $b$:
$$(a + b)^2 = a^2 + 2ab + b^2.$$
Each term has a name worth knowing:
$a^2$ — the first square term, the square of the first quantity.
$2ab$ — the middle term (or cross term), twice the product of the two quantities.
$b^2$ — the second square term, the square of the second quantity.
The companion identity is $(a - b)^2 = a^2 - 2ab + b^2$ — same shape, with the middle term flipped to minus. The two are mirror images, and confusing them is a common slip.
How Is the (a + b)² Formula Derived?
Two proofs are worth seeing — one algebraic, one geometric. They explain the same fact from different directions, and together they make the $2ab$ impossible to forget.
Algebraic proof — just multiply it out. A square means "times itself," so $(a + b)^2 = (a + b)(a + b)$. Expand using the distributive property (the same move as FOIL):
$$(a + b)(a + b) = a\cdot a + a\cdot b + b\cdot a + b\cdot b = a^2 + ab + ba + b^2.$$
Since $ab = ba$, the two middle terms combine:
$$= a^2 + 2ab + b^2.$$
The $2ab$ is not an extra rule — it is the two cross-products $ab$ and $ba$ that the distribution always produces.
Geometric proof — the area of a square. Draw a square whose side is $(a + b)$. Its total area is $(a + b)^2$. Now cut it with one horizontal and one vertical line at the point that splits each side into a length $a$ and a length $b$. The square falls into four pieces:
one $a \times a$ square, area $a^2$,
one $b \times b$ square, area $b^2$,
two $a \times b$ rectangles, area $ab$ each, totaling $2ab$.
The pieces must add up to the whole:
$$(a + b)^2 = a^2 + 2ab + b^2.$$
The two $ab$ rectangles are the middle term. Once a student has seen the square cut into four pieces, the dropped-$2ab$ mistake tends to disappear — the term has a shape, not just a position in a formula.
How Is (a + b)² Different from a² + b²?
This is the distinction the whole topic turns on. They are not equal:
$$(a + b)^2 = a^2 + 2ab + b^2, \qquad a^2 + b^2 = (a + b)^2 - 2ab.$$
$(a + b)^2$ means "add first, then square." $a^2 + b^2$ means "square each, then add." The gap between them is exactly $2ab$. A quick numeric check settles it: with $a = 3$, $b = 4$, $(3 + 4)^2 = 49$ but $3^2 + 4^2 = 25$ — a difference of $24 = 2(3)(4)$. The relationship $a^2 + b^2 = (a+b)^2 - 2ab$ is itself a useful tool — see our a² + b² formula page for how it solves problems where only the sum and product are known.
Examples of the (a + b)² Formula
Example 1
Expand $(x + 5)^2$.
Apply the formula with $a = x$, $b = 5$:
$$(x + 5)^2 = x^2 + 2(x)(5) + 5^2 = x^2 + 10x + 25.$$
Final answer: $x^2 + 10x + 25$.
Example 2
Expand $(3x + 4)^2$.
A tempting shortcut is to square each term and add — to write $(3x)^2 + 4^2 = 9x^2 + 16$.
Wrong path. Writing $(3x + 4)^2 = 9x^2 + 16$. Check it at $x = 1$: the real value is $(3 + 4)^2 = 49$, but $9 + 16 = 25$. The shortcut is off by $24$ — and $24 = 2(3)(4)$, the exact middle term that was dropped. Squaring a sum is not the same as summing the squares.
The break. The error treats squaring as if it distributed over addition: $(p + q)^2 \neq p^2 + q^2$. The distribution produces two cross-terms, and they don't vanish.
The rescue. Use the full formula with $a = 3x$, $b = 4$:
$$(3x + 4)^2 = (3x)^2 + 2(3x)(4) + 4^2 = 9x^2 + 24x + 16.$$
Final answer: $9x^2 + 24x + 16$. (Check at $x = 1$: $9 + 24 + 16 = 49 = 7^2$. ✓)
Example 3
Use the formula to compute $52^2$ mentally.
Write $52 = 50 + 2$ and apply the identity with $a = 50$, $b = 2$:
$$52^2 = (50 + 2)^2 = 50^2 + 2(50)(2) + 2^2 = 2500 + 200 + 4 = 2704.$$
Final answer: $52^2 = 2704$ — no long multiplication needed.
Example 4
Expand $(2x + 3y)^2$.
With $a = 2x$, $b = 3y$:
$$(2x + 3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2 = 4x^2 + 12xy + 9y^2.$$
Final answer: $4x^2 + 12xy + 9y^2$.
Example 5
Given $a + b = 7$ and $ab = 10$, find $a^2 + b^2$.
Rearrange the identity to isolate the sum of squares:
$$a^2 + b^2 = (a + b)^2 - 2ab = 7^2 - 2(10) = 49 - 20 = 29.$$
Final answer: $a^2 + b^2 = 29$ — found without ever solving for $a$ or $b$ individually.
Example 6
Factor $x^2 + 14x + 49$ back into a whole square.
Recognise the pattern $a^2 + 2ab + b^2$. Here $a^2 = x^2$ so $a = x$; $b^2 = 49$ so $b = 7$; and the middle term should be $2ab = 2(x)(7) = 14x$ — which matches. So it is a perfect square:
$$x^2 + 14x + 49 = (x + 7)^2.$$
Final answer: $(x + 7)^2$. (The middle-term check is what confirms the factoring; without it, $x^2 + 14x + 49$ might be mistaken for a non-square trinomial.)
Where the (a + b)² Formula Shows Up
This identity is one of the first algebraic tools that keeps paying off long after Class 8.
Completing the square. Solving any quadratic $ax^2 + bx + c = 0$ by completing the square rebuilds a perfect-square trinomial — running Example 6 in reverse. The quadratic formula itself is derived this way.
Mental arithmetic. Squaring numbers near a round figure ($52^2$, $98^2 = (100 - 2)^2$) is instant with the identity, a trick rooted in Vedic mathematics and still taught for fast computation.
Statistics — variance. The shortcut formula for variance, $\text{Var}(X) = E[X^2] - (E[X])^2$, is a direct cousin of the $a^2 + b^2 = (a+b)^2 - 2ab$ rearrangement; the cross-term is what separates "mean of the squares" from "square of the mean."
Geometry and physics. Any area, energy, or distance expression involving a squared sum — kinetic-energy terms, the expansion of $(\vec{u} + \vec{v})\cdot(\vec{u} + \vec{v})$ — carries the $2ab$ cross term, which in vectors becomes the dot-product interaction term.
For Class 8 to Class 10 students, the immediate use is expansion and factoring — but completing the square and the variance shortcut mean this one identity reappears in algebra, calculus prep, and statistics.
Tripping Points to Avoid
Mistake 1: Dropping the middle term
Where it slips in: Writing $(a + b)^2 = a^2 + b^2$, treating squaring as if it distributed over addition.
Don't do this: Square each term separately and stop. The two cross-products $ab + ba = 2ab$ are part of the expansion, not optional.
The correct way: Always write all three terms, $a^2 + 2ab + b^2$. In the McKinney TX Grade 8 cohort, this is the single most-repeated error on the first identities worksheet — it appears in close to half of first attempts, and it is the rusher archetype's signature: straight to "square both," no middle step.
Mistake 2: Confusing (a + b)² with (a − b)²
Where it slips in: Putting a minus on the middle term of a sum, or a plus on the middle term of a difference.
Don't do this: Assume the middle sign is always positive (or always negative). It tracks the sign inside the bracket.
The correct way: $(a + b)^2 = a^2 + 2ab + b^2$ (plus middle); $(a - b)^2 = a^2 - 2ab + b^2$ (minus middle). The first and last terms are positive either way — only the middle term carries the sign.
Mistake 3: Misreading the cross term when terms have coefficients
Where it slips in: Expanding $(3x + 4)^2$ and writing the middle term as $2(3)(4) = 24$ instead of $2(3x)(4) = 24x$ — forgetting the variable.
Don't do this: Take $2ab$ from the bare coefficients and lose the variable that belongs in $a$.
The correct way: Treat the whole term as $a$. For $(3x + 4)^2$, $a = 3x$, so $2ab = 2(3x)(4) = 24x$. The second-guesser archetype gets the structure right, then "simplifies away" the variable because the all-numbers version felt cleaner.
Conclusion
The a plus b whole square formula is $(a + b)^2 = a^2 + 2ab + b^2$ — two squares plus twice the product.
The $2ab$ middle term comes from the two cross-products of the distribution, shown geometrically as two equal rectangles in a square of side $a + b$.
$(a + b)^2$ is not $a^2 + b^2$; they differ by exactly $2ab$.
The most common errors are dropping the middle term, confusing it with $(a - b)^2$, and losing the variable in the cross term.
The identity powers completing the square, fast mental arithmetic, the variance shortcut, and vector dot-product expansions.
Practice These Before Moving On
Expand $(2x + 7)^2$.
Compute $103^2$ using the formula.
Given $a + b = 9$ and $ab = 14$, find $a^2 + b^2$.
If Problem 1 gave you $4x^2 + 49$, return to Mistake 1 — the $28x$ middle term is missing. For the related cube identities, see a³ + b³ formula.
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