Equations and Inequalities

Formulas (4)

• Expanding brackets and the special products · Special products
  $$(a \pm b)^2 = a^2 \pm 2ab + b^2 \qquad (a+b)(a-b) = a^2 - b^2$$
• Factorising: common factors, grouping and trinomials · Monic trinomial
  $$x^2 + sx + p = (x + m)(x + n) \quad\text{where}\quad m + n = s,\; mn = p$$
• Simplifying and combining algebraic fractions · Adding fractions
  $$\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$$
• Polynomial division and the factor theorem · Remainder theorem
  $$p(x) = (x - a)\,q(x) + p(a)$$

Watch out for (4)

• The square of a sum is not the sum of the squares
  → Expanding brackets and the special products
• A sum of two squares does not factorise
  → Factorising: common factors, grouping and trinomials
• Cancel factors, never terms
  → Simplifying and combining algebraic fractions
• For the factor x + a, test x = minus a
  → Polynomial division and the factor theorem

Formulas (2)

• Linear equations and equations with a parameter · Solution of ax = b
  $$ax = b,\; a \ne 0 \;\Rightarrow\; x = \frac{b}{a}$$
• Systems of two linear equations · Two equations in two unknowns
  $$\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$$

Watch out for (2)

• Zero coefficient: check the right-hand side
  → Linear equations and equations with a parameter
• Proportional coefficients do not always mean no solution
  → Systems of two linear equations

Formulas (2)

• Solving quadratic equations and the discriminant · Quadratic formula
  $$x = \frac{-b \pm \sqrt{\Delta}}{2a}, \qquad \Delta = b^2 - 4ac$$
• Sum and product of the roots · Sum and product of the roots
  $$x_1 + x_2 = -\frac{b}{a} \qquad x_1 x_2 = \frac{c}{a}$$

Watch out for (2)

• Never divide both sides by x
  → Solving quadratic equations and the discriminant
• The sum of the roots is minus b over a
  → Sum and product of the roots

Formulas (3)

• Linear inequalities and systems of inequalities · Multiplying by a negative number
  $$a < b \;\text{and}\; c < 0 \;\Rightarrow\; ac > bc$$
• Quadratic inequalities with a sign chart · Sign of a quadratic with a > 0 and roots x₁ < x₂
  $$a(x - x_1)(x - x_2) < 0 \;\text{for}\; x_1 < x < x_2, \qquad > 0 \;\text{for}\; x < x_1 \;\text{or}\; x > x_2$$
• Inequalities with absolute value · Absolute value inequalities (k > 0)
  $$|A| < k \;\Leftrightarrow\; -k < A < k \qquad |A| > k \;\Leftrightarrow\; A < -k \;\text{or}\; A > k$$

Watch out for (4)

• Dividing by a negative reverses the sign
  → Linear inequalities and systems of inequalities
• x² < 16 is not x < 4
  → Quadratic inequalities with a sign chart
• Do not multiply by a denominator of unknown sign
  → Fractional inequalities: signs of numerator and denominator
• Greater than gives two pieces, not one
  → Inequalities with absolute value

Formulas (3)

• Equations with the unknown in a denominator · Clearing the denominator
  $$\frac{P(x)}{Q(x)} = R(x) \;\Rightarrow\; P(x) = R(x)\,Q(x), \quad Q(x) \ne 0$$
• Equations and inequalities with square roots · Squaring a root equation
  $$\sqrt{A} = B \;\Rightarrow\; A = B^2 \quad\text{(keep only solutions with}\; B \ge 0)$$
• Exponential and logarithmic equations · Matching exponents or logs
  $$a^{f(x)} = a^{g(x)} \;\Rightarrow\; f(x) = g(x) \qquad \log_a f(x) = \log_a g(x) \;\Rightarrow\; f(x) = g(x) > 0$$

Watch out for (4)

• An excluded value can appear as an answer
  → Equations with the unknown in a denominator
• Squaring can add false solutions
  → Equations and inequalities with square roots
• A root inequality has a domain
  → Equations and inequalities with square roots
• Check the domain of every logarithm
  → Exponential and logarithmic equations