Lesson 1 · from §42 of the 1910 text
The Square of a Binomial
Multiplying a + b by itself gives a formula worth memorizing: (a + b)^2 = a^2 + 2ab + b^2.
I. The square of the sum of two terms is the sum of their squares plus twice their product.
Likewise (a - b)^2 = a^2 - 2ab + b^2.
II. The square of the difference of two terms is the sum of their squares minus twice their product.
These make mental arithmetic easy: 16^2 = (10 + 6)^2 = 100 + 36 + 2 \cdot 10 \cdot 6 = 256, and 49^2 = (50 - 1)^2 = 2500 + 1 - 100 = 2401.
Quick check
(x - 3)^2
Working backward
Since a^2 + 2ab + b^2 is (a + b)^2, dividing it by a + b gives a + b at once. A trinomial like that, two squares plus or minus twice their product, is a perfect square: x^2 - 10x + 25 = (x - 5)^2.
Quick check
Write y^2 + 8y + 16 as the square of a binomial.
Bigger "terms"
Sometimes a whole group counts as one term. In (a + b) - 3(a + 2b) + 4(a + 3b) the three pieces are called terms, and the expression is a trinomial. So to square x + y + b, treat x + y as one term: [(x + y) + b]^2 = (x + y)^2 + 2b(x + y) + b^2 = x^2 + 2xy + y^2 + 2bx + 2by + b^2.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Square each, using rule I or II.
16^2, using (10 + 6)^2
49^2, using (50 - 1)^2
15^2
17^2
18^2
19^2
(x + y)^2
(x - c)^2
(x + 1)^2
(a + 2)^2
(b^2 + 3)^2
(c - 4)^2
(2c^2 + d)^2
(3x - y)^2
(x + 2y)^2
(a - 4b)^2
(y^5 - 2)^2
(2a + 1)^2
(3a^2 + 2)^2
(4a + 3)^2
(2a^2 - 3b)^2
(3c^2 - 4a^2)^2
(2x^3 - 6)^2
(3xy + y^2)^2
(7x^2 - 3y^4)^2
(8y^3 + 2bx)^2
(6xy - 2a^2)^2
(4xy - 2x^2)^2
(3x^4 - 5x^6)^2
(2x^2 - 6x^3y)^2
41^2
92^2
101^2
202^2
1001^2
Divide in your head.
\dfrac{a^2 + 2ab + b^2}{a + b}
\dfrac{a^2 - 2ac + c^2}{a - c}
\dfrac{a^2 - 4a + 4}{a - 2}
\dfrac{9x^2 - 6x + 1}{3x - 1}
\dfrac{y^2 + 40y + 400}{20 + y}
\dfrac{9x^2 - 12xy + 4y^2}{2y - 3x}
\dfrac{4a^2 - 20a + 25}{-5 + 2a}
\dfrac{4x^2 - 12xy + 9y^2}{3y - 2x}
Find a binomial divisor of each trinomial.
x^2 - 2xy + y^2
a^2 + 2a + 1
a^2 + 4a + 4
4x^2 + 4x + 1
16x^2 - 8x + 1
16x^2 - 8xy + y^2
9x^4 - 6x^2 + 1
16y^6 - 40y^3 + 25
Write each as the product of two binomials.
c^2 + 2cd + d^2
a^2 - 2a + 1
x^2 - 10x + 25
4a^2 - 4a + 1
4x^2 + 12x + 9
9a^2b^2 - 6ab + 1
16x^4y^2 - 24x^2ya + 9a^2
25a^4b^6 - 60a^2b^3c^2 + 36c^4
Square each trinomial by treating two of its terms as one.
(a + b + c)^2
(a + b - c)^2
(a + b - 1)^2
(a + b + 1)^2
(a - b + c)^2
(a - b - c)^2
(a + 2b - 3)^2
(a - 2b + c)^2
(2a + b - 4c)^2
(3a - 2b + 5c)^2
(2a - 3b - 4c)^2
(4a - 1 + 2ab)^2