Quadratic Equations

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Concept 1: Introduction

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants and a cannot be zero.

Key Definitions

• Quadratic Equation: An equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a cannot be zero.
• Roots: The solutions to a quadratic equation, which are the values of x that make the equation true.
• Coefficient: A constant that multiplies a variable in a polynomial, e.g., a and b are coefficients in ax^2 + bx + c.

Types of Quadratic Equations

• Monic Quadratic Equation: A quadratic equation in which the coefficient of the squared variable is 1, i.e., x^2 + bx + c = 0.
• Non-Monic Quadratic Equation: A quadratic equation in which the coefficient of the squared variable is not 1.

Key Formulas

• Sum and Product of Roots: If α and β are the roots of the quadratic equation ax^2 + bx + c = 0, then: α + β = -b/a αβ = c/a
• Quadratic Formula: If α and β are the roots of the quadratic equation ax^2 + bx + c = 0, then: α = (-b ± √(b^2 - 4ac)) / 2a β = (-b ± √(b^2 - 4ac)) / 2a
• Discriminant: The expression b^2 - 4ac under the square root in the quadratic formula.
• Nature of Roots: The nature of the roots of a quadratic equation can be determined using the discriminant:
  • If b^2 - 4ac > 0, the roots are real and distinct.
  • If b^2 - 4ac = 0, the roots are real and equal.
  • If b^2 - 4ac < 0, the roots are complex.

Key Theorems

• Factorization Theorem: A quadratic equation can be factored into the product of two binomial factors if its roots are rational.
• Rational Root Theorem: If a quadratic equation has a rational root, then it must be a factor of the constant term.

Examples

• Solve the quadratic equation x^2 + 5x + 6 = 0 using the quadratic formula. x = (-5 ± √(5^2 - 4(1)(6))) / 2(1) x = (-5 ± √(25 - 24)) / 2 x = (-5 ± √1) / 2 x = (-5 ± 1) / 2 x = -3 or x = -2

Practice Questions

• Solve the quadratic equation 2x^2 + 7x + 3 = 0 using the quadratic formula.
• Find the roots of the quadratic equation x^2 - 4x + 4 = 0.
• Determine the nature of the roots of the quadratic equation x^2 + 9x + 20 = 0.