Algebra Study Notes: Linear & Quadratic Equations

Algebra Study Notes

Linear Equations

• Definition: An equation of the form ( ax + b = 0 ) where ( a ) and ( b ) are constants.
• Solutions: Solve for ( x ) by isolating it: ( x = -\frac{b}{a} ).
• Graph Representation: A straight line on a coordinate plane.

Quadratic Functions

• Standard Form: ( f(x) = ax^2 + bx + c ).
• Vertex Form: ( f(x) = a(x-h)^2 + k ) where ( (h, k) ) is the vertex.
• Graphing: Parabola shape; opens upwards if ( a > 0 ), downwards if ( a < 0 ).
• Roots: Found using the quadratic formula: ( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} ).

Factorization Techniques

• Common Methods:
  • Grouping: Group terms and factor out common factors.
  • Quadratics: Use patterns (e.g., ( a^2 - b^2 = (a - b)(a + b) )).
  • Trial and Error: Finding factors that multiply to give ( ac ) and add to give ( b ).

Graphing

• Axes: Horizontal (x-axis) and vertical (y-axis).
• Scaling: Properly scale each axis for accurate representation.
• Key Points: Identify intercepts, vertices, and asymptotes for functions.
• Tools: Linear equations are graphed as lines, quadratic functions as parabolas.

Matrix

• Definition: A rectangular array of numbers arranged in rows and columns.
• Operations:
  • Addition/Subtraction: Combine corresponding elements.
  • Multiplication: Dot product of rows and columns.
• Applications: Solve systems of equations and transformations in geometry.

Determinants

• Definition: A scalar value that describes certain properties of a matrix.
• Calculation:
  • For ( 2 \times 2 ) matrix: ( \text{det}(A) = ad - bc ) where ( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} ).
• Properties:
  • A matrix is invertible if its determinant is non-zero.
  • Determines the area/volume in geometric interpretations.

Discriminants

• Definition: Part of the quadratic formula ( b^2 - 4ac ).
• Interpretation:
  • If ( > 0 ): Two distinct real roots.
  • If ( = 0 ): One real root (repeated).
  • If ( < 0 ): No real roots (complex roots).

Solving Systems of Equations

• Methods:
  • Substitution: Solve one equation for a variable and substitute into the other.
  • Elimination: Add or subtract equations to eliminate a variable.
  • Graphical Method: Graph both equations; the intersection point(s) give the solution.

Solving Systems of Inequalities

• Graphical Representation:
  • Each inequality represents a region on the graph.
  • The solution is the intersection of these regions.
• Testing Points: Choose test points in each region to determine which satisfies the inequalities.
• Common Forms: ( ax + by < c ), ( ax + by \geq c ), etc.

Linear Equations

• A linear equation has the form (ax + b = 0), where (a) and ( b ) are constants.
• The solution for (x) is found by isolating it: (x = -\frac{b}{a}).
• Linear equations are represented graphically as straight lines on a coordinate plane.

Quadratic Functions

• The standard form of a quadratic function is: (f(x) = ax^2 + bx + c), where (a), (b), and (c) are constants.
• The vertex form of a quadratic function is: (f(x) = a(x - h)^2 + k), where ( (h, k) ) is the vertex of the parabola.
• Quadratic functions graph as parabolas. The parabola opens upwards if (a > 0) and downwards if (a < 0).
• The roots of a quadratic equation are the solutions to the equation (f(x)=0). They can be found using the quadratic formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).

Factorization Techniques

• Factorisation simplifies expressions and is used to solve equations.
• Common methods include:
  • Grouping: Grouping terms together to factor out common factors.
  • Quadratics: Applying patterns like (a^2 - b^2 = (a - b)(a + b)).
  • Trial and Error: Finding factors that multiply to give (ac) and add to give (b).

Graphing

• The x-axis is horizontal and the y-axis is vertical.
• Proper scaling is important for accurate representation of the graph.
• Key points to identify for graphs include intercepts, vertices, and asymptotes.
• Linear equations are graphed as straight lines, while quadratic functions create parabolas.

Matrix

• A matrix is a rectangular array of numbers organized into rows and columns.
• Matrix operations include:
  • Addition/Subtraction: Adding or subtracting corresponding elements of two matrices.
  • Multiplication: Calculating the dot product of rows and columns.
• Matrices are used in solving systems of equations and geometric transformations.

Determinants

• A determinant is a scalar value that describes certain properties of a matrix.
• For a (2 \times 2) matrix (A = \begin{pmatrix} a & b \ c & d \end{pmatrix}), the determinant is calculated as: (det(A) = ad - bc).
• Determinants are used to determine whether a matrix is invertible (non-zero determinant) and are related to areas and volumes in geometric applications.

Discriminants

• The discriminant of a quadratic equation is the part of the quadratic formula: ( b^2 - 4ac ).
• The discriminant helps determine the nature of the roots:
  • If (b^2 - 4ac > 0 ), there are two distinct real roots.
  • If (b^2 - 4ac = 0), there is one real root (repeated).
  • If (b^2 - 4ac < 0 ), there are no real roots (complex roots).

Solving Systems of Equations

• Systems of equations involve multiple equations with multiple variables.
• Common methods for solving include:
  • Substitution: Solving one equation for a variable and substituting that value into the other equation.
  • Elimination: Adding or subtracting equations to eliminate a variable.
  • Graphical Method: Graphing both equations and finding the point(s) where they intersect, which represents the solution.

Solving Systems of Inequalities

• Systems of inequalities involve multiple inequalities with multiple variables.
• Each inequality represents a region on a graph.
• The solution is the intersection of all the regions that represent the inequalities.
• Test points are chosen inside each region to determine which satisfies the inequalities.
• Common forms of linear inequalities include: ( ax + by < c ), ( ax + by \geq c ), etc.