Mathematics · Foundational · 35 min · Chapter 2 of 28

Algebra

Letters stand for numbers. Equations are balances. The quadratic formula closes the story of every second-degree polynomial.

Read first: Arithmetic and number sense

The idea

Algebra begins the moment a symbol is allowed to stand for a number you do not yet know. The rules do not change: you may add the same thing to both sides, multiply both sides by the same nonzero number, and expand with the distributive law. What changes is the goal. You are rearranging until the unknown is isolated, and every step is reversible so you have not added or lost solutions.

A linear equation ax + b = 0, with a ≠ 0, has the single solution x = −b/a. Two such equations, in two unknowns, usually meet at one point. They fail to meet when the lines are parallel (same slope, different intercept) and they coincide when the equations are scalar multiples of each other. That trichotomy — one solution, none, or infinitely many — is the preview of linear algebra.

A quadratic ax² + bx + c = 0 is the first equation whose solution is not a finite combination of the coefficients with plus, minus, times, and divide. Completing the square produces the quadratic formula. The discriminant D = b² − 4ac tells the character of the roots over the reals: two distinct when D > 0, one repeated when D = 0, and none when D < 0. Over the complex numbers those last two roots exist and are conjugates.

Hold onto this

Solving an equation means producing an equivalent equation until the unknown stands alone.
A linear equation has at most one solution. A line is its graph.
Every quadratic ax² + bx + c factors in the complexes, and the quadratic formula finds the roots.
Logarithms turn products into sums because they are the inverses of exponential functions.

Formulas

Linear equation

a x + b = 0 ⟹ x = - \frac{b}{a} (a \neq = 0)

One step: move the constant, then divide by the coefficient of x.

a: Coefficient of the unknown, not zero
b: Constant term

Slope of a line

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Rise over run. Positive slope climbs as x grows; zero slope is horizontal.

(x₁, y₁), (x₂, y₂): Two distinct points on the line

Quadratic formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

The two roots of ax² + bx + c = 0. The expression under the radical is the discriminant.

a, b, c: Coefficients, a ≠ 0
±: Compute both the plus and the minus

Logarithm of a product

lo g_{b} (x y) = lo g_{b} x + lo g_{b} y

A logarithm converts multiplication into addition. It is the inverse of bˣ.

b: Base, positive and not 1
x, y: Positive real numbers

Square of a sum

(a + b)^{2} = a^{2} + 2 ab + b^{2}

The middle term is easy to forget and is the whole reason completing the square works.

2ab: Twice the product of the two terms

Worked examples

1. Two roots, both integers

Solve x² − 5x + 6 = 0.

Here a = 1, b = −5, c = 6.
Discriminant: (−5)² − 4(1)(6) = 25 − 24 = 1.
Roots: (5 ± 1) / 2.
Plus gives 6/2 = 3. Minus gives 4/2 = 2.

x = 3 or x = 2. Check: (x − 2)(x − 3) = x² − 5x + 6.

2. A logarithm undoes a power

Solve 2ˣ = 40. Leave the answer as a logarithm, then approximate.

Apply log base 2, or any common log, to both sides.
x = log₂(40) = ln 40 / ln 2.
ln 40 ≈ 3.689 and ln 2 ≈ 0.693, so x ≈ 5.32.
Check: 2⁵ = 32 and 2⁶ = 64, so a value a little above 5 is reasonable.

x = ln 40 / ln 2 ≈ 5.32.

Try these

Solve 3(x − 2) = 2x + 5.

For which c does x² + 2x + c = 0 have exactly one real root?

Common mistakes

Dividing both sides by an expression that might be zero can delete a root. Factor instead.
√(b² − 4ac) is a single nonnegative number. The ± outside it is what produces two roots.
(a + b)² is not a² + b². The cross term 2ab is not optional.