1
Q
p(x) = (2m - 1)x² + (m + 1)x + (m - 4) can be written as a perfect square. Find the value of m.
A
- Find the value of a, b and c
- Write the equation in the form b² - 4ac = 0
- Simplify the equation into quadratic form
- Solve equation with a calculator to find the values of m
- Eliminate one solution, as the expression is a perfect square
2
Q
By factorizing, find the expression in terms of p for the difference between the roots of the equation (px)² + 4px - 12 = 0
A
- Find the value of a, b and c
- Write the equation using the quadratic formula x = -b ± √b² - 4ac / 2a
- Simplify the equation to remove the square root
- Solve for if the equation is positive and negative
3
Q
Show that the graph of the function y = (x - a)(x - b) - c², where c ≠ 0, crosses the x-axis at two distinct points.
A
- Expand brackets
- Factorise into the quadratic form
- Find the value of a, b and c
- Write the equation in the form b² - 4ac > 0
- Simplify the equation (add brackets and powers if necessary) to leave two terms
- Write the answer as ‘since (term 1) > 0 and (term 2) > 0, c ≠ 0
4
Q
- Powers
A
- When multiplying terms, add the powers
- When dividing terms, subtract the powers
- When raising a term with a power to another power, multiply the powers
5
Q
- Negative powers
A
Numbers
- With fractions, invert and raise each number to its positive power (simplify the fractions first if necessary)
Variables
- To change a - power to +, put the variable with the negative power on the bottom of the fraction
- Use brackets when more than 1 expression is being raised to a power
6
Q
- Surds
A
Surd form = Index form
ᵃ√xᵇ = x b/a
- Convert the expression into index form
- Use power laws to simplify the expression (split the terms under the square root if necessary).
7
Q
- Roots, powers and coefficients
A
- When raising a number/variable with a power to another power, use brackets, then simplify to remove the brackets
8
Q
- Powers (base is a number)
A
- Rewrite the expression with the smallest base possible
2. Use the power laws to simplify the expression
9
Q
- Logarithm rules
A
- log a + log b = log ab
- log a - log b = log a/b
- n log a = log aⁿ
- Rewrite the expression using the logarithm of the smallest number possible
- Simplify using the logarithm rules
- logᵦ 1 = 0
- logᵦ b = 1
- logᵦ(mn) = logᵦm + logᵦn
- logᵦ(m/n) = logᵦm - logᵦn
- logᵦmⁿ = nlogᵦm
- logᵦ(1/x) = -logᵦx
10
Q
- Solving logarithm equations - Basic
A
- Rewrite logᵦ y = x as y = bˣ
2. Use calculator to find y
11
Q
- Solving logarithm equations - Finding the base or exponent
A
- Rewrite logᵦ y = x as y = bˣ
2. Figure out value of b, then check on calculator
12
Q
- Solving logarithm equations - Finding the exponent with messy numbers
A
- Take the log of both sides
- Use log rules to rearrange it > x log a = log b
- Divide other side of equation by log a to remove it > x = log b / log a
- Use calculator to find the value of x
13
Q
- Applications of exponential equations
A
1. Rewrite in the form A = Prⁿ, where P = The starting value r = The rate of change n = The number of time periods (often years) over which change occurs A = The final amount
- Rates of change
- No change r = 1
- Increase r > 1
- Decrease r < 1
- Add/subtract percentage to/from 100 to determine rate of change. If it is an increase, add. If it is a decrease, subtract. Use (100 ± x / 100) - If asked to calculate n, take the log of both sides
14
Q
- Fractions - Multiplying and dividing
A
- Multiplying fractions
a/b x c/d = ac/bd
- Dividing fractions
(Invert the second fraction and multiply)
a/b / c/d = ad/bc
15
Q
- Fractions - Adding and subtracting
A
- With same denominator
a/b ± c/b = a±c/b
- With different denominator
Multiply both the numerator and denominators by a number or variable that will create equal denominators
a/b ± c/d = ad/bd ± bc/bd = ad ± bc/bd
16
Q
- Fractions - Equations
A
Multiply every term on both sides by the lowest common multiple of the denominators, then simplify
17
Q
- Fractions - Inequations
A
Multiply every term on both sides by the lowest common multiple of the denominators, then simplify
- If you multiply or divide the equation by a negative number, you must reverse the sign
18
Q
- Polynomials - Simplifying rational quadratic expressions
A
Factorize the top and bottom of the fraction, then cancel out common factors
19
Q
- Polynomials - Solving quadratic equations
A
- Arrange into ax^2 + bx + c = 0
2. Enter a, b and c values into the calculator to solve for x values
20
Q
- Polynomials - Solving quadratic inequations
A
- Arrange into ax² + bx + c = 0
- Enter a, b and c values into the calculator to solve for x values
- Substitute a value that is between the solutions to see if the inequation is true or not
- If it is true, write the answer as a domain. If it is false, write the x values as less than the bottom number, and more than the top number.
21
Q
- Polynomials - Solving rational quadratic equations
A
- If possible, factorise and cancel out
- Remove the bottom line of fractions by multiplying to the other side of the equation
- Solve
- If two solutions are found at the end, substitute them into the factorised original equation and solve for each. If one produces a denominator of 0, it is not a solution.
22
Q
- Roots of equations - Calculating value of discriminant and number of roots
A
Note - If there is no a, b or c value, it is = to 0
- Arrange into ax² + bx + c = 0
- Find the value of a, b and c
- Write the equation in the form △ = b² - 4ac
- If answer > 0, there are two, real, distinct roots. If answer = 0, there is one, real root. If answer < 0, there are no real roots.
23
Q
- Roots of equations - Calculating values of a, b or c, when given the number of roots
A
- Arrange into ax² + bx + c = 0
- Find the value of a, b and c
- Write the equation in the form △ = b² - 4ac
- Make equation < = > to 0 depending on number of roots given
- Solve to find a, b or c value
24
Q
- Roots of equations - Solving when substitution is required
A
Note - √(ax + b)² = (ax + b) and (aⁿ)² = (a²)ⁿ = a²ⁿ
- Let variable ‘p’ = to a value/equation that can be substituted with the a and b values of equation
- Substitue p with a and b values of equation
- Mak equation < = > to 0 depending on number of roots given
- Solve to find value(s) of p
- Substitue each of the p values into the value/equation for p to solve for the x value(s)
- Check x answers work
25
13. Forming and solving quadratic equations - Given information about the equation
1. Read what the question asks for and call it x
- If there are two things, call the smaller one x and the larger one (x + _)
2. If resulting in 2 answers, test and reject one, then explain why it was rejected
- Call even numbers 2x
- Call odd numbers 2x + 1
- Call consecutive numbers x, x + 1, x + 2
- Call consecutive even numbers 2x, 2x + 2, 2x + 4
- Call consecutive odd numbers 2x + 1, 2x + 3, 2x + 5
- With age questions, give youngest age x
26
13. Forming and solving quadratic equations - Given information about the equation of graph to find equation of curve
1. Parabola
x-intercepts -
y = ±a (x±b)(x±c)
vertex -
y = ±a (x±b)^2 ±c
27
14. Rearranging expressions - Where subject appears once
1. Get rid of fractions by multiplying the lowest common multiple of the denominators
2. Use normal equation solving rules
28
14. Rearranging expressions - Where subject appears more than once
1. Get rid of fractions by multiplying the lowest common multiple of the denominators
2. Collect x terms on the left and others on the right
3. Factorise the left side with x outside the brackets
4. Divide so x becomes the subject
29
14. Rearranging expressions - Where there is a root sign
1. Reorganise so root term is on the left
2. Square both sides
3. Reorganise so x becomes the subejct
