Quadratic Equations

Quadratic Equations

• Polynomial equations of the second degree.
• A quadratic equation's roots or solutions are the values of 'x' that satisfy the equation.
• Solutions to quadratic equations can be found through factoring, completing the square, the quadratic formula, or graphing.
• The general form is ax² + bx + c = 0, where a ≠ 0, and a, b, and c are coefficients while x is an unknown variable.

Factoring

• Factoring expresses the quadratic expression as a product of two linear factors.
• Roots of the equation are derived by setting each factor to zero and solving for 'x'.
• x² + 5x + 6 = (x + 2)(x + 3) is an example.
• For the example equation, the roots are x = -2 and x = -3.
• Most effective when the quadratic expression can be easily factored.

Completing the Square

• Involves manipulating the quadratic equation into a perfect square trinomial.
• For x² + bx + c = 0, add and subtract (b/2)² to complete the square: x² + bx + (b/2)² - (b/2)² + c = 0.
• Can be written as: (x + b/2)² = (b/2)² - c.
• Solving for x is done by taking the square root of both sides.
• Useful when the quadratic equation cannot be easily factored.

Quadratic Formula

• General solution for any quadratic equation is x = (-b ± √(b² - 4ac)) / (2a) given ax² + bx + c = 0.
• The discriminant, b² - 4ac, determines the nature of the roots.
  • Two distinct real roots exist if b² - 4ac > 0.
  • One real root (a repeated root) exists if b² - 4ac = 0.
  • Two complex roots exist if b² - 4ac < 0.

Graphing Quadratic Equations

• A parabola is the graph of a quadratic equation.
• The real roots of the equation are represented by the parabola's x-intercepts.
• The minimum or maximum value of the quadratic function is represented by the vertex of the parabola.
• The axis of symmetry is a vertical line through the vertex dividing the parabola into two symmetrical halves.

Linear Functions

• Can be represented by a straight line on a graph.
• General form: y = mx + c, where:
  • 'y' is the dependent variable.
  • 'x' is the independent variable.
  • 'm' is the slope or gradient of the line.
  • 'c' is the y-intercept, where the line crosses the y-axis.
• The slope 'm' is the rate of change of 'y' with respect to 'x.'
• A positive slope indicates an increasing line, a negative slope indicates a decreasing line, and a 0 slope indicates a horizontal line.
• Linear functions can be represented graphically, algebraically, or in tabular form.

Slope-Intercept Form

• The slope and y-intercept of the line are directly shown in the form y = mx + c.
• Useful for quickly graphing a linear function or identifying its key properties.

Point-Slope Form

• y - y₁ = m(x - x₁) is the point-slope form, where (x₁, y₁) is a point on the line and 'm' is the slope.
• Useful when you know a point on the line and the slope.

Graphing Linear Functions

• Plot at least two points and then draw a straight line through them.
• The y-intercept (0, c) is often a convenient starting point.
• Use the slope to find additional points by moving 'm' units vertically for every 1 unit horizontally.

Finding the Equation of a Line

• Given two points (x₁, y₁) and (x₂, y₂), the slope of the line is m = (y₂ - y₁) / (x₂ - x₁).
• Use the point-slope form to find the equation then convert to slope-intercept form if desired.

Trigonometric Ratios

• Relate the angles of a right triangle to the ratios of its sides.
• Sine (sin), cosine (cos), and tangent (tan) are the primary ratios.
  • sin(θ) = Opposite / Hypotenuse
  • cos(θ) = Adjacent / Hypotenuse
  • tan(θ) = Opposite / Adjacent
• 'Opposite' is the length of the side opposite to angle θ.
• 'Adjacent' is the length of the side adjacent to angle θ.
• 'Hypotenuse' is the longest side, opposite the right angle.
• Used to solve problems involving angles and side lengths in triangles.

Reciprocal Trigonometric Ratios

• Cosecant (csc), secant (sec), and cotangent (cot) are the reciprocal trigonometric ratios.
  • csc(θ) = 1 / sin(θ) = Hypotenuse / Opposite
  • sec(θ) = 1 / cos(θ) = Hypotenuse / Adjacent
  • cot(θ) = 1 / tan(θ) = Adjacent / Opposite

Applications of Trigonometric Ratios

• Used to find unknown side lengths or angles in right triangles.
• Used in fields like navigation, surveying, engineering, and physics.

Angle of Elevation and Depression

• The angle of elevation is the angle between the horizontal line and the line of sight to an object above the horizontal.
• The angle of depression is the angle between the horizontal line and the line of sight to an object below the horizontal.

Solving Right Triangles

• Solving involves finding all unknown side lengths and angles.
• Use trigonometric ratios, the Pythagorean theorem (a² + b² = c²), and the sum of angles in a triangle is 180°.

Simultaneous Equations

• A set of two or more equations containing two or more variables.
• A solution is a set of values for the variables that satisfies all equations simultaneously.

Methods for Solving Simultaneous Equations

• Methods include substitution, elimination (addition or subtraction), and graphing.

Substitution

• Solve one equation for one variable in terms of the other(s).
• Substitute the expression into the other equation(s) to eliminate that variable.
• Solve the resulting equation(s) for the remaining variable(s).
• Substitute the values back into the original equation(s) to find the values of the other variables.

Elimination

• Multiply equations by a constant so that the coefficients of one variable are equal or opposite.
• Add or subtract the equations to eliminate that variable.
• Solve the resulting equation for the remaining variable.
• Substitute the value back into one of the original equations to find the value of the eliminated variable.

Graphing

• Graph each equation on the same coordinate plane.
• The point(s) where the graphs intersect represent the solution(s).
• Graphing is most effective for systems of two equations with two variables.

Applications of Simultaneous Equations

• Used to model and solve real-world problems involving multiple variables and relationships.
• Examples include mixture problems, rate problems, and geometry problems.