Maths

• What is the first function machine's output equation?

  $y =$$x^2 +$$6$
• What is the second function machine's output equation?

  $y =$$(x - 3)^2$
• What is the range of input values for which A is less than B?

  $x < \frac{1}{2}$
• What is the first step to solve the function machine problem?

  Set up equations based on outputs
• What does it mean when the output is the same as the input in a function machine?

  The input equals the output
• How do you find the original input in the second function machine question?

  Set up an equation and solve
• What method is used to solve the simultaneous equations in the last function machine question?

  Substitution or elimination method
• What are the key steps to solve a Venn diagram probability question?

  • Identify total number of people
  • Define categories and overlaps
  • Fill in known values
  • Calculate remaining values
  • Find probabilities based on totals
• What are the steps to solve quadratic inequalities?

  1. Set up the inequality
  2. Rearrange to one side
  3. Factor or expand as needed
  4. Solve for the variable
  5. Determine the range of values
• What are the characteristics of function machines in algebra?

  • Input is transformed to output
  • Can involve multiple operations
  • May require solving for unknowns
  • Can be represented with equations
• What are the strategies to avoid mistakes in calculations?

  • Take your time
  • Double-check each step
  • Explain your reasoning aloud
  • Write down all workings
• Why can simultaneous equations appear in different contexts?

  They can be used to solve various problems
• What does the same coefficient in front of b allow us to do?

  Subtract the equations directly
• What is the value of b after substituting a back into the equation?

  b = 10
• What is the value of a after solving the equations?

  a = 5
• Why is trial and error not recommended for this problem?

  It can be tricky with multiple equations
• What is the growth rate of bacteria in flask A?

  50% per day
• How does the population growth in flask A form a geometric progression?

  It multiplies by 1.5 each day
• What is the iterative formula for the bacteria population?

  P_{N + 1} = 1.5 P_N
• How many bacteria are in flask A at the start of day 6?

  75
• How many bacteria are in flask A at the start of day 10?

  384
• What is the formula to solve a quadratic equation?

  $x =$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• What is the formula for completing the square?

  $x^2 +$$bx +$$c =$$(x + \frac{b}{2})^2 - \frac{b^2}{4} +$$c$
• Why is completing the square a useful method for solving quadratic equations?

  It allows you to rewrite the equation in the form of a perfect square, making it easier to find the solutions
• How can you use the quadratic formula or completing the square to determine if a quadratic equation has no real roots?

  If the discriminant (b^2 - 4ac) is negative, then the equation has no real roots
• What are the steps to solve a quadratic equation using the quadratic formula?

  1. Identify the coefficients a, b, and c in the equation ax^2 + bx + c = 0
  2. Plug the coefficients into the quadratic formula:

  $x =$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

  1. Simplify the expression and solve for the two values of x
• How can you use the properties of geometric progressions to model the growth of a bacterial population?

  • The population grows exponentially at a constant rate
  • The population at the start of each day forms a geometric progression
  • The common ratio is the growth rate (e.g. 1.5 for 50% growth per day)
  • Can use the formula P_n = P_0 * r^n to find the population on any given day
• How do the growth rates of 50% per day for Flask A and 30% per day for Flask B affect the relative sizes of the bacterial populations over time?

  The population in Flask A will grow much faster and become significantly larger than Flask B over time
• If the population of bacteria in Flask A on the 10th day is 10 times the population on the 6th day, what is the value of the common ratio r?

  $r =$$5.06$
• What are the key differences between the growth of the bacterial populations in Flask A and Flask B?

  • Flask A has a growth rate of 50% per day
  • Flask B has a growth rate of 30% per day
  • The population in Flask A will grow much faster and become significantly larger than Flask B over time
  • The growth curve for Flask A will be much steeper than the more gradual growth curve for Flask B
• How can you use the quadratic formula to determine if a quadratic equation has no real roots?

  If the discriminant (b^2 - 4ac) is negative, then the equation has no real roots
• What is the formula for finding the solution to a quadratic equation to 6 decimal places?

  Keep iterating the solution using the formula until the 6th decimal place does not change
• How can you use the completing the square method to determine if a quadratic equation has no real roots?

  1. Rewrite the equation in the form x^2 + bx + c = 0
  2. Apply the completing the square formula:

  x^2 + bx + c = (x + b/2)^2 - b^2/4 + c

  1. If the expression inside the square (b^2/4 - c) is negative, then the equation has no real roots
• How do the different methods of solving quadratic equations (quadratic formula, completing the square) compare in terms of their advantages and disadvantages?

  The quadratic formula is more general but can be more computationally intensive, while completing the square provides more insight into the structure of the equation
• How can you use the properties of geometric progressions to model the growth of a bacterial population over time?

  • The population grows exponentially at a constant rate
  • The population at the start of each day forms a geometric progression
  • Can use the formula P_n = P_0 * r^n to find the population on any given day, where:
  • P_n is the population on day n
  • P_0 is the initial population
  • r is the common ratio (growth rate)
  • n is the number of days
• How can you use the properties of geometric progressions to compare the growth of bacterial populations in two different flasks?

  • If Flask A has a growth rate of 50% per day and Flask B has a 30% growth rate per day:
  • The population in Flask A will grow much faster and become significantly larger than Flask B over time
  • The growth curve for Flask A will be much steeper than the more gradual growth curve for Flask B
  • Can use the formula P_n = P_0 * r^n to calculate and compare the populations on any given day
• What is the formula for the area of a circle with radius r?

  $A =$$\pi r^2$
• What are the key differences between using the quadratic formula versus completing the square to solve a quadratic equation?

  The quadratic formula is more general but can be more computationally intensive, while completing the square provides more insight into the structure of the equation
• If a quadratic equation has the form x^2 + bx + c = 0, how can you use the completing the square method to determine if it has no real roots?

  Rewrite the equation as (x + b/2)^2 - b^2/4 + c = 0. If b^2/4 - c is negative, then the equation has no real roots.
• What are the advantages and disadvantages of using the quadratic formula versus completing the square to solve quadratic equations?

  Advantages of quadratic formula:

  • More general, can be applied to any quadratic equation
  • Provides the exact solutions

  
  

  Advantages of completing the square:

  • Provides more insight into the structure of the equation
  • Can be used to determine if an equation has no real roots

  
  

  Disadvantages:

  • Quadratic formula can be more computationally intensive
  • Completing the square requires more steps and manipulation of the equation