Lesson 1 · from §57 of the 1910 text
Solving Equations by Factoring
Some words
A term is rational if it can be built from the letters using only +, -, \times, \div, with no roots taken: 15x, \tfrac{a-b}{2}, and \sqrt{4a^2} are rational, while \sqrt{2x} and \sqrt{a - b} are not. A term is integral if it has no letter in a denominator.
The degree of an equation in one unknown is the highest power of the unknown. Degree 2 is quadratic (x^2 - 6x = 18), degree 3 is cubic, and degree 4 is biquadratic.
The key principle
If a product of factors is zero, one of the factors must be zero.
Example. Solve x^2 + 5x = 6. Transpose: x^2 + 5x - 6 = 0. Factor: (x - 1)(x + 6) = 0. If x - 1 = 0, then x = 1; if x + 6 = 0, then x = -6. Check: 1 + 5 = 6 and 36 - 30 = 6. ✓
A cubic. x^3 + x^2 = 4x + 4 becomes x^2(x + 1) - 4(x + 1) = 0, then (x + 1)(x + 2)(x - 2) = 0, with roots -1, -2, 2.
Rule. Transpose so the right side is zero. Factor the left side, set each factor containing the unknown equal to zero, and solve each.
An equation usually has as many distinct roots as its degree, and never more. Factors can repeat, though: x^2 - 6x + 9 = (x - 3)(x - 3) = 0 has the single root 3.
Never divide both sides by an expression containing the unknown, because you can lose roots that way.
Quick check
x^2 - 5x + 6 = 0
From the history of algebra
A famous fake proof. Let x = 1. Then x^2 = x, so x^2 - 1 = x - 1, and (x - 1)(x + 1) = x - 1. Divide by x - 1: x + 1 = 1, so 2 = 1! The error is dividing by x - 1, which is zero when x = 1. That's exactly why the rule above forbids dividing by an expression with the unknown in it.
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.
Find the roots of each quadratic. Type them separated by commas, like 3, -2.
x^2 - 9 = 0
x^2 = 16
x^2 - 3x = 0
x^2 = 7x
x^2 - 7x = -12
x^2 - x = 20
3x^2 - 15x = 0
5x^2 + 35x = 0
8 - 9x = -x^2
12x - 28 = -x^2
4x^2 = 16x
x^2 - a^2 = 0
x^2 = 9b^2
x^2 - 2ax + a^2 = 0
x^2 + 4b^2 = 4bx
x^2 + ax + 3x + 3a = 0
x^2 + bx = 4x + 4b
x^2 + ax + bx + ab = 0
4x^2 + 8x + 3 = 0
9x^2 = 3x + 2
16x^2 - 12x = 10
50x + 24 = 25x^2
x^2 + bx = 0
x^2 - ax - bx = 0
x^2 + 3x = ax
4x = x^2 + 4
Solve each cubic.
x^3 - 9x = 0
x^3 + x^2 = 4x + 4
x^3 - 5x^2 + 6x = 0
2x^3 - x^2 = 32x - 16
5x^2 + x^3 = 45 + 9x
Find the roots of each biquadratic.
x^4 - 5x^2 + 4 = 0
9 + x^4 = 10x^2
x^4 - 36x^2 = 0
x^4 + 15x^2 = 8x^3
x^4 - 2x^2 + 1 = 0