Lesson 7 · from §54 of the 1910 text
The Difference of Two Squares
The type form is a^2 - b^2 = (a + b)(a - b).
Rule. Take the square root of each term (treating both as positive). Their sum is one factor, and their difference is the other.
Keep going while a factor still splits: x^8 - y^4 = (x^4 + y^2)(x^4 - y^2) = (x^4 + y^2)(x^2 + y)(x^2 - y).
Quick check
x^4 - 16
Groups as terms
Either "square" can be a whole group: 4 - (a + b)^2 = [2 + (a + b)][2 - (a + b)] = (2 + a + b)(2 - a - b).
Four or six terms can often be rearranged into a difference of squares: a^2 + 2ab + b^2 - c^2 - 2cd - d^2 = (a + b)^2 - (c + d)^2 = (a + b + c + d)(a + b - c - d). Look for a perfect-square trinomial hiding inside.
Quick check
x^2 + 6x + 9 - y^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.
Factor completely.
k^2 - h^2
h^2 - 1
k^2 - 4
a^4 - x^2
1 - a^2
1 - 4a^2
4a^2 - 9
25 - x^2
h^4 - 81
16x^4 - 25
36c^8 - 49d^8
a^2b^4 - 121
100 - z^8
x^4 - y^4
64h^6 - 169
a^8 - b^8
1 - 16x^4
16x^4 - y^8
x^16 - y^16
x^16 - 4y^16
144a^2b^4 - 9
196 - x^4y^6
225a^4 - 16b^8c^12
x^{2n} - y^2
x^2 - y^{4n}
a^{4n} - b^2
a^{6n} - b^{2n}
a^{4n} - b^{6n}
4x^{2n} - y^{4n}
9x^{2n} - 4y^{6n}
(a + b)^2 - 4
(x - y)^2 - 4c^2
(a - 2b)^2 - 9x^2
4(x + y)^2 - 1
25(3b - c)^2 - 64
4 - (a + b)^2
81 - (x - y)^2
4a^2 - (2x + y)^2
Factor as a difference of two squares, grouping terms where needed.
(a - b)^2 - 9x^2
16y^2 - (h + 2k)^2
25a^4 - (2b - 3c)^2
49x^6 - (7c - 2d)^2
1 - (5h - 3k)^2
(5x + 2y)^2 - (3x - 7y)^2
(a - b)^2 - (a + b)^2
(a - b)^2 - (x + y)^2
(a + 2b)^2 - (a - 3b)^2
(2a - 3b)^2 - (3a - 2b)^2
(5h + 3k)^2 - (2c - 9d)^2
(4b - a)^2 - (7a - 6b)^2
4(a - 2b)^2 - (2x - y)^2
9(a + b)^2 - (2a - 5b)^2
(a - 2b)^2 - 4(a + b)^2
y^2 - c^2 + x^2 - 2xy
x^2 + 2ax + a^2 - y^2
x^2 + 2x + 1 - 4z^2
c^2 - 2cd + d^2 - 16a^2
4x^2 - 12cx + 9c^2 - 25y^2
x^2 - y^2 - 2ax + a^2
k^2 - g^4 - 4kh + 4h^2
6xy - c^2 + 9x^2 + y^2
12ab - 4h^2 + 4b^2 + 9a^2
1 - 4ax + 4a^2x^2 - x^2
4c^2 - 20cd + 25d^2 - 9d^4
9d^2 - 25a^2 - 6cd + c^2
12ab + x^2 - 4a^2 - 9b^2
a^2 - b^2 - 2bc - c^2
x^2 - a^2 - 2ac - c^2
y^2 - b^2 + 4bc - 4c^2
2bc - c^2 - b^2 + a^4
9x^2 - 4y^2 - a^2 - 4ay
6x + 9y^2 - 9 - x^2
4bc + 1 - 4c^2 - b^2
4bc - 4b^2 + 4x^6 - c^2
x^2 - a^2 + y^2 - 4 - 2xy + 4a
a^2 + 2ab + b^2 - c^2 - 2cd - d^2
9k^2 - 6hk + h^2 - 4c^2 - 4cd - d^2
x^2 - 1 + y^2 - a^2 + 2xy + 2a
1 + 2bc + 2a - c^2 - b^2 + a^2
Review: factor completely.
x^3 - x
x^4 - 2x^2 + 1
x^8 - 2x^4 + 1
x^5 - 8x^3 + 16x
x^5 - x
x^10 - x^2
x^4 - 10x^2 + 9
x^4 - 13x^2 + 36
3a^2x^4 - 12a^4x^2 + 12a^6
18a^2x^2 - 24a^2x - 10a^2
3x^4 - 15x^2 + 12
12a - 39ay - 51ay^2
4n^6 + 48n^2 - 28n^4
16x^4 + 8x^2 - 3
a^3 - a + a^2b - b
x^3 - x^2 - 4x + 4
3a^3 + 3a^2 - 27a - 27
2a^3b + 3a^2b - 8ab - 12b
4a^2 - a^4 + 81 + 10a^2x - 36a - 25x^2
12cd^3 - 6a^3x - a^6 + 4c^2 + 9d^6 - 9x^2