MATH

• When analysing markets, a range of assumptions are made about the rationality of economic agents involved in the transactions
• The Wealth of Nations was written

  1776
• Rational


  (in classical economic theory) economic agents are able to consider the outcome of their choices and recognise the net benefits of each one
• Consumers act rationally by


  Maximising their utility
• Producers act rationally by


  Selling goods/services in a way that maximises their profits
• Workers act rationally by


  Balancing welfare at work with consideration of both pay and benefits
• Governments act rationally by


  Placing the interests of the people they serve first in order to maximise their welfare
• Rationality in classical economic theory is a flawed assumption as people usually don't act rationally
• A firm increases advertising

  Demand curve shifts right
• Marginal utility


  The additional utility (satisfaction) gained from the consumption of an additional product
• If you add up marginal utility for each unit you get total utility
• Solving quadratic equations by extracting square roots

  1. Write the equation in the form x2 = k

  2. Apply the following properties:

  a. If k > 0, then x2 = k has two real solutions or roots: x = ±√k

  b. If k = 0, then x2 = k has one real solution or root: x = 0

  c. If k < 0, then x2 = k has no real solutions or roots
• Quadratic equations of the form ax2 + c = 0

  • x2 + 6 = 0
  • 2x2 + 12 = 0

  25x2 - 4 = 0
• Solving quadratic equations by extracting square roots (examples)

  Example 1: Solve x2 - 25 = 0

  Example 2: Solve m2 = 0

  Example 3: Solve y2 + 64 = 0

  Example 4: Solve (x - 5)2 - 81 = 0
• Solving quadratic equations by factoring

  Transform the quadratic equation into standard form if necessary

  b. Factor the quadratic expression

  c. Apply the zero product property by setting each factor of the quadratic expression equal to 0

  d. Solve each resulting equation

  e. Check the values of the variable obtained by substituting each in the original equation
• Solving quadratic equations by factoring

  • Example 1: Solve x2 + 12x = -11

  Example 2: Solve 16x2 - 9 = 0

  Example 3: Solve 3y2 + 24y = 0
• Solving quadratic equations by completing the square


  Divide both sides of the equation by a then simplify

  2. Write the equation such that the terms with variables are on the left side of the equation and the constant term is on the right side

  3. Add the square of one-half of the coefficient of x on both sides of the resulting equation. The left side of the equation becomes a perfect square trinomial

  4. Express the perfect square trinomial on the left side of the equation as a square of a binomial

  5. Solve the resulting quadratic equation by extracting the square root

  6. Solve the resulting linear equations

  7. Check the solutions obtained against the original equation
• Solving quadratic equations by completing the square

  • Example 1: Solve 2x2 + 8x - 24 = 0
• Solving quadratic equation 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0 by completing the square

  1. Divide both sides by a

  2. Write equation with variables on left, constant on right

  3. Add square of half coefficient of x to both sides

  4. Express left side as perfect square trinomial

  5. Solve resulting perfect square equation

  6. Solve resulting linear equations

  7. Check solutions against original equation
• Solving quadratic equations using the quadratic formula

  1. Write equation in standard form

  2. Determine values of a, b, c

  3. Substitute a, b, c into quadratic formula

  4. Simplify result

  5. Check solutions against original equation
• Solve 2x^2 = 1 - 2x


  1. Write the equation in standard form

  2. Determine the values of a, b, and c

  3. Substitute the values of a, b, and c in the quadratic formula

  4. Simplify the result
• Quadratic formula


  x = -b±√(b^2-4ac)/(2a)
• x = 4
• x = -3
• 2x^2 + 2x - 1 = 0
• The equation 2x^2 + 2x - 1 = 0 has two irrational roots/solutions: x = -(1+√3)/2 or x = -(1-√3)/2
• 4m^2 - 10m = 0
• t^2 + 4t + 12 = 0
• 2x - 3 = 0
• 3p + 1 = 20
• y^2 + 3y = 2
• x^2 - 2x + 15 = 0
• 12w - 6 = 8
• 7 - 2r = r^2
• s^2 - 81 = 0
• 3x + 9x + 1 = x + 3x
• The length of a swimming pool is 8 m longer than its width and the area is 105 m^2
• Edna paid at least Php1,200 for a pair of pants and a blouse. The cost of the pair of pants is Php600 more than the cost of the blouse
• The area of a rectangular plot is 528 m^2. The length of the plot (in meters) is one more than twice its width
• The product of two consecutive positive integers is 306