Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Mathematics > Algebra > Flashcards

Algebra Flashcards

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
  1. Find the value of a, b and c
  2. Write the equation in the form b² - 4ac = 0
  3. Simplify the equation into quadratic form
  4. Solve equation with a calculator to find the values of m
  5. Eliminate one solution, as the expression is a perfect square
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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
  1. Find the value of a, b and c
  2. Write the equation using the quadratic formula x = -b ± √b² - 4ac / 2a
  3. Simplify the equation to remove the square root
  4. Solve for if the equation is positive and negative
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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
  1. Expand brackets
  2. Factorise into the quadratic form
  3. Find the value of a, b and c
  4. Write the equation in the form b² - 4ac > 0
  5. Simplify the equation (add brackets and powers if necessary) to leave two terms
  6. Write the answer as ‘since (term 1) > 0 and (term 2) > 0, c ≠ 0
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q
  1. 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
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q
  1. 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
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q
  1. Surds
A

Surd form = Index form
ᵃ√xᵇ = x b/a

  1. Convert the expression into index form
  2. Use power laws to simplify the expression (split the terms under the square root if necessary).
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q
  1. Roots, powers and coefficients
A
  • When raising a number/variable with a power to another power, use brackets, then simplify to remove the brackets
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q
  1. Powers (base is a number)
A
  1. Rewrite the expression with the smallest base possible

2. Use the power laws to simplify the expression

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q
  1. Logarithm rules
A
  1. log a + log b = log ab
  2. log a - log b = log a/b
  3. 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
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q
  1. Solving logarithm equations - Basic
A
  1. Rewrite logᵦ y = x as y = bˣ

2. Use calculator to find y

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q
  1. Solving logarithm equations - Finding the base or exponent
A
  1. Rewrite logᵦ y = x as y = bˣ

2. Figure out value of b, then check on calculator

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q
  1. Solving logarithm equations - Finding the exponent with messy numbers
A
  1. Take the log of both sides
  2. Use log rules to rearrange it > x log a = log b
  3. Divide other side of equation by log a to remove it > x = log b / log a
  4. Use calculator to find the value of x
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q
  1. 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
  1. 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)
  2. If asked to calculate n, take the log of both sides
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q
  1. 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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q
  1. 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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q
  1. Fractions - Equations
A

Multiply every term on both sides by the lowest common multiple of the denominators, then simplify

17
Q
  1. 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
  1. Polynomials - Simplifying rational quadratic expressions
A

Factorize the top and bottom of the fraction, then cancel out common factors

19
Q
  1. Polynomials - Solving quadratic equations
A
  1. Arrange into ax^2 + bx + c = 0

2. Enter a, b and c values into the calculator to solve for x values

20
Q
  1. Polynomials - Solving quadratic inequations
A
  1. Arrange into ax² + bx + c = 0
  2. Enter a, b and c values into the calculator to solve for x values
  3. Substitute a value that is between the solutions to see if the inequation is true or not
  4. 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
  1. Polynomials - Solving rational quadratic equations
A
  1. If possible, factorise and cancel out
  2. Remove the bottom line of fractions by multiplying to the other side of the equation
  3. Solve
  4. 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
  1. 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

  1. Arrange into ax² + bx + c = 0
  2. Find the value of a, b and c
  3. Write the equation in the form △ = b² - 4ac
  4. 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
  1. Roots of equations - Calculating values of a, b or c, when given the number of roots
A
  1. Arrange into ax² + bx + c = 0
  2. Find the value of a, b and c
  3. Write the equation in the form △ = b² - 4ac
  4. Make equation < = > to 0 depending on number of roots given
  5. Solve to find a, b or c value
24
Q
  1. Roots of equations - Solving when substitution is required
A

Note - √(ax + b)² = (ax + b) and (aⁿ)² = (a²)ⁿ = a²ⁿ

  1. Let variable ‘p’ = to a value/equation that can be substituted with the a and b values of equation
  2. Substitue p with a and b values of equation
  3. Mak equation < = > to 0 depending on number of roots given
  4. Solve to find value(s) of p
  5. Substitue each of the p values into the value/equation for p to solve for the x value(s)
  6. Check x answers work
25
Q
  1. Forming and solving quadratic equations - Given information about the equation
A
  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
Q
  1. Forming and solving quadratic equations - Given information about the equation of graph to find equation of curve
A
  1. Parabola
    x-intercepts -
    y = ±a (x±b)(x±c)

vertex -
y = ±a (x±b)^2 ±c

27
Q
  1. Rearranging expressions - Where subject appears once
A
  1. Get rid of fractions by multiplying the lowest common multiple of the denominators
  2. Use normal equation solving rules
28
Q
  1. Rearranging expressions - Where subject appears more than once
A
  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
Q
  1. Rearranging expressions - Where there is a root sign
A
  1. Reorganise so root term is on the left
  2. Square both sides
  3. Reorganise so x becomes the subejct

Mathematics (12 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions