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.

Explore essential algebra concepts including linear equations, quadratic functions, and factorization techniques. This quiz will help reinforce your understanding of graphing and solving equations effectively. Get ready to test your knowledge on these fundamental algebra topics!