Algebra Notes
Basic types of equations
| Equation Type | Equation Example |
|---|---|
| $(5)$ | Monomial (one term) with a degree of 0 (constant). |
| $(3x + 2)$ | Binomial (two terms) with a degree of 1 (linear). |
| $(x^2 - 4x + 7)$ | Trinomial (three terms) with a degree of 2 (quadratic). |
| $(x^3 + 2x^2 - 5x - 6)$ | Polynomial with a degree of 3 (cubic). |
| $ $ | $ $ |
Definition of equation forms
| Factored Form | Expanded Form |
|---|---|
| $(x + 4)(x - 6)$ | $(x^2 - 2x - 24)$ |
| $(x-2)(4x+5)$ | $(4x^2-3x-10)$ |
| $(x + 4)(x - 6)$ | $(x^2 - 2x - 24)$ |
| $x(4x-3)(2x-1)$ | $(8x^3+2x^2-3x)$ |
| $(7x-3)(7x+3)$ | $(49x^2-9)$ |
| $x(x-6)(x+6)$ | $(x^3-36x)$ |
| $ $ | $ $ |
Un-simplified vs Simplified equations
| Un-simplified | Simplified |
|---|---|
| $\sqrt{x^{13}}$ | $x^6\sqrt{x}$ |
| $\sqrt[3]{27w^3}$ | $3w$ |
| $\sqrt{32}-\sqrt{8}$ | $2\sqrt{2}$ |
| $ $ | $ $ |
Exploring Radical Equations
| Given | Radical Notation | Simplified |
|---|---|---|
| $9^{\frac{1}{2}}$ | $\sqrt{9}$ | $3$ |
| $ $ | $ $ | $ $ |
Properties of Quadratic Equations
Graphing Quadratic Equations
Finding Max/Min or Vertex of Parabolas
Aka: finding the maximum or minimum value of a quadradic equation.
When the equation is in standard form:
① ax² + bx + c = 0.
② The vertex's $x$ value is: $\frac{-b}{2a}$.
③ To get the $y$ value, plug in $x$.
When the equation is in Vertex form:
① $y = a(x-h)^2 + k$ An example being: $y = 2(x-1)^2 + 2$
② The vertex is at: $(h,k)$ -> Example vertex is at: $(1, 3)$
IMPORTANT:
① In example: $y = 2(x+3)^2 +4$
② Re-write as: $y = 2(x-(-3))^2 + 4$
③ Following $(h,k)$, vertex is at: $(-3, 4)$
What is the discriminant?
In the Quadratic Formula: $x = \frac{-b ± \sqrt{b^2 -4ac}}{2a}$
The quantity $b^2 -4ac$ is called the discriminant.
The discriminant is frequently denoted by a Δ where Δ = b² - 4ac.
The discriminant reveals the nature of roots:
Δ > 0 & Square number: The equation has two distinct real and rational solutions/roots.
Δ > 0 & Non-square number: The equation has two distinct real and irrational solutions/roots.
Δ = 0: The equation has exactly one real and rational solution (a repeated root).
Δ < 0: The equation has two complex (non-real) solutions.
Analyzing: 2x²+9x-5
The equation intersects with the X axis two times.
2x²+9x-5 factored is: $(2x-1)(x+5)$
Meaning there are roots at $x=-5$ and $x=\frac{1}{2}$.
Aka these are points: $(-5,0)$ and $(\frac{1}{2},0)$.
Vertex form: $2(x+\frac{9}{4})^2 - \frac{121}{8}$.
Equation has a vertex at: $(-\frac{9}{4},-\frac{121}{8})$.
Analyzing: 4x²-20x+25
The equation intersects with the X axis one time. Parabola opens upward, a>0.
4x²-20x+25 factored is: $(2x-5)^2$
Meaning there is one root at $x=\frac{5}{2}$.
Aka these are points: $(\frac{5}{2},0)$.
The vertex's $x$ value is: $\frac{-b}{2a}=\frac{-(-20)}{(2*4)}=\frac{20}{8}=\frac{5}{2}$.
The vertex's $y$ is: $y=4*\frac{25}{4}-20*\frac{5}{2}+25=25-50+25=0$.
Equation has a vertex at: $(\frac{5}{2},0)$.
Analyzing: 3x²+2x+9
The equation does not cross the X axis. Parabola opens upward, a>0.
Two complex roots: $x=-\frac{1}{3}±\frac{i\sqrt{26}}{3}$
Vertex form: $3(x+\frac{1}{3})^2+\frac{26}{3}$
Vertex at $(-\frac{1}{3},\frac{26}{3})$
Dictionary
| Term | Definition |
|---|---|
| Relation | A relation is any set of ordered pairs typically enclosed in curly braces "{}". In text, relations look like: {(x , y), ..} or {(0,-3),(1,-2),(2,-1),(3,0),..} |
| Domain | The set of all x-values of the relation. See "Relation". Domain: {0,1,2,3}. |
| Range | The set of all y-values of the relation. See "Relation". Range: {-3,-2,-1,0}. |
| Factoring | The process of splitting a product into its factors. $8*7=56$. $8$ and $7$ are factors. $56$ is a product. |
| Greatest Common Factor (GCF) | The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions. |
| Polynomial | A polynomial is a mathematical expression consisting of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. They can be classified by the number of terms (monomial, binomial, trinomial, etc.) or by the highest degree of the variable. |
| Variables | Symbols (usually letters) that represent unknown values. |
| Coefficients | Numerical values that multiply the variables. |
| Exponents | Non-negative integer powers to which the variables are raised. |
| Terms | Individual components of a polynomial separated by addition or subtraction. |
| "Degree" or Degree of the polynomial | The highest power of the variable in a polynomial. |
| Vertex | In the context of quadratic equations, the vertex refers to the highest or lowest point on the parabola, depending on whether it opens upward or downward. |
| $y=(5)$ | Monomial (one term) with a degree of 0 (constant). |
| $(3x + 2)$ | Binomial (two terms) with a degree of 1 (linear). |
| $(x^2 - 4x + 7)$ | Trinomial (three terms) with a degree of 2 (quadratic). |
| $(x^3 + 2x^2 - 5x - 6)$ | Polynomial with a degree of 3 (cubic). |
| "Quadratic" or Quadratic Equation | A quadratic equation is a polynomial equation of the second degree. See "Trinomial". |
| General Form or "Standard Form" | The standard way to write a quadratic equation is ax² + bx + c = 0. |
| Parabola | The graph of a quadratic equation is a parabola. |
| $(x + 4)(x - 6)$ | Distributes to: $(x^2 - 2x - 24)$ |
| Discriminant | The value that helps determine the nature of the roots (solutions) of a quadratic equation. Typically: Δ, sometimes "d". Δ = b² - 4ac |
| Rational Equation | Rational numbers have decimal representations that either terminate or repeat. Examples: 34, 1, 1/2, -3/4, 5, 0.25. Not a √. |
| Irrational Equation | Irrational numbers have non-terminating, non-repeating decimal representations. Examples: √2, π, e. Usually a √. |