Quadratic Equations: Formulas and Roots

• Quadratic equations and statistics are important mathematical concepts

Quadratic Equations

• A quadratic equation represents a second-degree polynomial equation.
• It is generally expressed as ax² + bx + c = 0, where a ≠ 0.

Solving Quadratic Equations

• Factoring involves expressing the quadratic expression as a product of two linear factors.
• The quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a)
• Completing the Square: Transform the equation into the form (x + h)² = k to solve for x.

The Discriminant

• The discriminant (Δ = b² - 4ac) determines the nature of the roots of the quadratic equation.
• If Δ > 0, the equation has two distinct real roots.
• If Δ = 0, the equation has one real root (a repeated root).
• If Δ < 0, the equation has two complex roots.

Sum and Product of Roots

• For a quadratic equation ax² + bx + c = 0, the sum of the roots equals -b/a.
• The product of the roots equals c/a.

Forming Quadratic Equations

• Given the roots α and β, the quadratic equation can be formed as x² - (α + β)x + αβ = 0.

Applications of Quadratic Equations

• Used in physics to describe projectile motion.
• Used in engineering to design curves and structures.
• Used in computer graphics for rendering shapes.

Statistics

• Statistics involves collecting, organizing, analyzing, interpreting, and presenting data.

Types of Data

• Qualitative data (categorical) represents characteristics or attributes like colors, names, or labels.
• Quantitative data (numerical) signifies measurable quantities, whether discrete (countable) or continuous (measurable).

Measures of Central Tendency

• Mean: The average of a set of numbers, which is derived by summing the values and dividing by the number of values.
• Median: The middle value in a sorted set of numbers.
• Mode: The most frequently appearing value in a set of numbers.

Measures of Dispersion

• Range: Obtained by subtracting the minimum value from the maximum value in a dataset.
• Variance: The average of the squared differences from the mean.
• Standard Deviation: The square root of the variance, indicating the spread of data around the mean.

Frequency Distribution

• A frequency distribution is a table or chart displaying the frequency of different values in a dataset.
• Relative frequency is the frequency of a value divided by the total number of values.

Histograms

• A histogram is a graphical representation of a frequency distribution, using bars to represent the frequency of data within intervals.

Probability

• Probability is the measure of the likelihood that an event will occur.
• Probability ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.

Conditional Probability

• Conditional probability is the probability of an event occurring given that another event has already occurred.

Regression Analysis

• Regression analysis is a statistical technique used to model the relationship between one dependent variable and one or more independent variables.
• Linear regression models this relationship using a linear equation.

Correlation

• Correlation measures the strength and direction of the linear relationship between two variables.
• The correlation coefficient, ranging from -1 to 1, indicates the strength and direction of this relationship.
• 1 indicates a perfect positive correlation
• -1 indicates a perfect negative correlation
• 0 indicates no correlation.