All of ALGEBRA in under 1 hour (ultimate study guide)

Main ideas / lessons (top-level)

• Algebra uses variables to represent unknown quantities, and most core skills involve rewriting expressions and solving for values of variables.
• The video walks through 10 must-know algebra topics, moving from simplifying expressions to equations/inequalities, then functions, graphs, systems of equations, exponents, polynomials, factoring, and finally quadratic functions/equations.
• Across topics, the recurring theme is to apply rules consistently to maintain equivalence (e.g., balance method for equations/inequalities) and look for structures (like terms, like variables/exponents, intercepts, vertex, zero product rule).

1) Simplifying algebraic expressions

Key definitions

• Term: a number, a variable, or a product of numbers/variables.
  • Examples: 7, x, 5x^2, -3xy^5
• Algebraic expression: multiple terms combined using operations (addition/subtraction, etc.).
  • Example: 5x^2 - 3x + 1

Like terms (what they are)

Like terms have:

• the exact same variables, and
• the exact same exponents.

Notes:

• Coefficients can differ and they can still be like terms (e.g., 5x^2 and -5x^2).
• If the variable differs (e.g., 5x vs 5y), they are not like terms.

Method to simplify (collect like terms)

Steps:

1. Rearrange/group like terms next to each other.
2. Keep the sign attached to each term (the sign to the left “belongs” to the term).
3. For each like-term group, add/subtract coefficients while keeping the variable part the same.

Example pattern:

• Combine 3x^2 - 5x^2 → (3 - 5)x^2 = -2x^2
• Combine -2x + 4x → (-2 + 4)x = 2x
• Constants combine only with other constants (e.g., 10 + 3, etc.).

Distributive property (expanding)

A monomial a multiplied by a bracket expression (x + y) becomes:

• a(x + y) = ax + ay

Workflow in the video:

• Multiply the outside term by each inside term.
• Then collect like terms again to simplify.

2) Solving linear equations

Definition / goal

• Solve an equation means finding the variable value(s) that make the equation true.

Balance method (inverse operations)

• Maintain equality by doing the same operation to both sides.
• Isolate the variable using inverse operations.

Order rule used:

• Follow BEDMAS in reverse (inverse sequence):
  1. do addition/subtraction first,
  2. then multiplication/division,
  3. then exponents,
  4. then inside brackets.

Example workflow shown

For 2x + 5 = 11:

1. Subtract 5 from both sides → 2x = 6
2. Divide by 2 → x = 3

Verification:

• Substitute the solution back into the original equation to confirm both sides match.

More complex linear equation approach

• Move all variable terms to one side.
• Move all constant terms to the other side.
• When moving a term across the equal sign, you change its sign.

3) Inequalities

What an inequality is

• Compares two quantities using: >, <, ≥, ≤.

Goal

• Find all x-values that satisfy the inequality.
• Unlike equations (single answer), inequalities give a range.

Solving rule

• Same as linear equations except:
  • If you multiply or divide by a negative number, you must flip the inequality sign.

Representation of solutions

• Number line
  • Open circle for > or < (not included)
  • Closed circle for ≥ or ≤ (included)
  • Shade the solution set
• Interval notation
  • Parentheses () when endpoints are not included
  • Square brackets [] when endpoints are included

4) Functions and function notation

Function definition

• A relationship where each input maps to exactly one output.

Function notation

• f(x) means the output of the function named f when the input is x.
• It is not multiplication of f and x.

Evaluating a function (table/points)

• Replace x with a chosen input and compute the output.
• Example: if f(x) = 2x^2 + 1, then:
  • f(2) = 2(2^2) + 1 = 9
  • Plot the point (2, 9)

Vertical line test (to confirm function)

• If any vertical line hits the graph more than once, it’s not a function.
• If every vertical line hits it at most once, it is a function.

5) Graphing linear functions (slope-intercept form)

Form

• y = mx + b

Meanings

• m = slope
  • rate of change: change in y / change in x
  • also phrased as rise / run
• b = y-intercept
  • where the line crosses the y-axis: point (0, b)

Graphing process

1. Plot the y-intercept.
2. Use the slope to find additional points.

Example shown: For y = (-1/3)x + 5:

• Plot (0, 5)
• Slope -1/3:
  • rise -1 (down 1), run 3 (right 3)
• Draw the line through the points.

6) Systems of equations

Definition

• Find values of variables (e.g., x and y) that satisfy all equations in the system.

Three main methods

Method 1: Graphing

Steps:

1. Rewrite each equation in slope-intercept form (y = mx + b)
2. Graph both lines
3. The intersection point gives the solution (x, y)

Verification:

• Substitute the solution into both equations.

Method 2: Substitution

Steps:

1. Isolate one variable in one equation
2. Substitute that expression into the other equation
3. Solve the resulting single-variable equation
4. Substitute back to find the second variable

Workflow example:

• Express x in terms of y, plug into equation 1, solve for y, then solve for x.

Method 3: Elimination

Steps:

1. Stack equations so like terms line up
2. Multiply one (or both) equations so coefficients of one variable match (same absolute value)
3. Add/subtract equations to eliminate that variable
4. Solve for the remaining variable
5. Substitute back

7) Exponents

Definition of power

• a^n means repeated multiplication:
  • a multiplied by itself n times.

Examples of expansion

• 9^2 = 9 * 9 = 81
• (-5)^2 = (-5) * (-5) = 25

Sign note emphasized:

• If the negative is not inside parentheses, it can behave differently (example indicates it can lead to -25).

Exponent rules (power laws)

• Product rule: a^m * a^n = a^(m+n)
• Quotient rule: a^m / a^n = a^(m-n) (same base)
• Power of a power: (a^m)^n = a^(m*n)

8) Multiplying polynomials

Core rule

• Multiply every term in the first polynomial by every term in the second.

Part A: binomial × binomial (FOIL)

FOIL:

• First
• Outside
• Inside
• Last

After expanding, combine like terms.

Part B: trinomial × binomial

• Multiply all terms with all terms.
• Then simplify by combining like terms.

9) Factoring

What factoring is

• The reverse of expansion:
  • rewrite an expression as a product of simpler expressions (factors).

Three types covered

1) Common factoring

Steps:

1. Find the greatest common factor (GCF) across all terms
2. Factor it out by dividing each term by the GCF and writing the result in parentheses

2) Quadratic trinomial factoring (leading coefficient = 1)

• Form: x^2 + bx + c

Steps:

1. Find numbers m and n such that:
   • m + n = b
   • m * n = c
2. Then:
   • x^2 + bx + c = (x + m)(x + n)

3) Quadratic trinomial factoring (leading coefficient ≠ 1)

For ax^2 + bx + c with a ≠ 1:

Steps:

1. Find two numbers whose:
   • product is a*c
   • sum is b
2. Split the middle term bx into two terms using those numbers
3. Factor by grouping
   • factor out common factors from the first two terms and from the last two terms
4. Then factor out the shared binomial

10) Quadratic functions and equations (Mushroom Academy section)

Quadratic function basics

• A quadratic is a polynomial where the highest power of x is 2.
• The graph is a parabola with a vertex:
  • opens up → vertex is the minimum
  • opens down → vertex is the maximum

Vertex coordinates (method shown)

• From standard form ax^2 + bx + c:
  • x_vertex = -b / (2a)
• Then compute:
  • y_vertex by substituting x_vertex into the quadratic.

Graphing using the vertex and x-intercepts

• x-intercepts occur where y = 0.
• To find x-intercepts:
  • set the quadratic equal to 0
  • convert to factored form if possible
  • use the zero product rule:
    • if (factor1)(factor2) = 0, then factor1 = 0 or factor2 = 0

Solving quadratics (two methods shown)

Method A: Factoring + zero product rule

1. Factor the quadratic
2. Set each factor equal to zero
3. Solve the resulting linear equations

Method B: Quadratic formula

• Formula:
  • x = (-b ± sqrt(b^2 - 4ac)) / (2a)
• Steps shown:
  • substitute a, b, c
  • simplify inside the square root (discriminant)
  • compute both solutions using ±

Speakers / sources featured

• Jensen May