Section 7

Equations

fundamentalviz

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

Solution of a first-degree equation.

fundamental

A = B C \Rightarrow B = \frac{A}{C}, C = \frac{A}{B}

Isolates a variable by applying equivalent operations.

fundamentalviz

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

Solves every quadratic equation with a nonzero.

The term under the radical is the discriminant. Its sign determines whether the roots are distinct real, repeated real, or complex conjugates.

fundamentalviz

Δ = b^{2} - 4 a c

Determines the nature of the roots of a real quadratic.

fundamentalviz

x^{2} + b x = (x + \frac{b}{2})^{2} - \frac{b ^{2}}{4}

Rewrites a quadratic as a square plus a constant.

fundamental

\frac{P ( x )}{Q ( x )} = \frac{R ( x )}{S ( x )} \Rightarrow P (x) S (x) = R (x) Q (x)

Clears denominators while respecting domain restrictions.

fundamental

f (x) = g (x) \Rightarrow f (x) = g (x)^{2}

Isolating and raising powers can solve radicals, but solutions must be checked.

fundamentalviz

∣ f (x) ∣ = a ⟺ f (x) = a o f (x) = - a (a \geq 0)

Turns an absolute-value equation into two cases.