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Quadratic Equations

Definition

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term that is squared (exponent of 2). The standard form is:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0.

Prerequisites

To solve and understand quadratic equations, you need proficiency in:

  1. Polynomials (operations)
  2. Factoring
  3. Exponents & Radicals (especially square roots)
  4. Basic Equation Solving
  5. Order of Operations

Learning Objectives

After mastering this topic, you should be able to:

  1. Identify quadratic equations and write them in standard form.
  2. Solve quadratic equations by factoring.
  3. Solve quadratic equations using the square root property (when b = 0).
  4. Solve quadratic equations by completing the square.
  5. Solve quadratic equations using the quadratic formula.
  6. Understand and use the discriminant (b² - 4ac) to determine the number and type of solutions (roots).
  7. Graph basic quadratic functions (parabolas) and identify key features (vertex, axis of symmetry, intercepts).
  8. Solve real-world problems modeled by quadratic equations.

Methods for Solving Quadratic Equations

1. Factoring

  • Set the equation to zero (standard form).
  • Factor the quadratic expression.
  • Set each factor equal to zero and solve for x (Zero Product Property).
  • Example: Solve x² + 5x + 6 = 0
    • Factor: (x + 2)(x + 3) = 0
    • Set factors to zero: x + 2 = 0 or x + 3 = 0
    • Solutions: x = -2 or x = -3

2. Square Root Property

  • Use when the equation is in the form ax² + c = 0 (no linear term).
  • Isolate the x² term.
  • Take the square root of both sides, remembering both positive and negative roots (±).
  • Example: Solve 2x² - 18 = 0
    • Isolate x²: 2x² = 18 -> x² = 9
    • Take square root: x = ±√9
    • Solutions: x = 3 or x = -3

3. Completing the Square

  • A method to rewrite the quadratic ax² + bx + c = 0 into the form (x + h)² = k.
  • Useful for deriving the quadratic formula and graphing.
  • Steps: Move constant term, divide by a (if a≠1), add (b/2a)² to both sides, factor the perfect square trinomial, solve using square root property.

4. Quadratic Formula

  • Solves any quadratic equation in standard form ax² + bx + c = 0.
  • Formula: x = [-b ± √(b² - 4ac)] / 2a
  • Example: Solve x² + 3x - 5 = 0
    • a=1, b=3, c=-5
    • x = [-3 ± √(3² - 4(1)(-5))] / 2(1)
    • x = [-3 ± √(9 + 20)] / 2
    • x = [-3 ± √29] / 2
    • Solutions: x = (-3 + √29)/2 or x = (-3 - √29)/2

The Discriminant (b² - 4ac)

  • Found inside the square root in the quadratic formula.
  • Determines the nature of the solutions:
    • If b² - 4ac > 0: Two distinct real solutions.
    • If b² - 4ac = 0: One real solution (a repeated root).
    • If b² - 4ac < 0: Two complex conjugate solutions (no real solutions).

Examples

Example 1: Solving by Factoring (High School)

Problem: Solve 2x² - 5x = 3. Solution: 1. Standard form: 2x² - 5x - 3 = 0 2. Factor: (2x + 1)(x - 3) = 0 3. Zero Product Property: 2x + 1 = 0 or x - 3 = 0 4. Solve: x = -1/2 or x = 3

Example 2: Using Quadratic Formula (High School)

Problem: Solve 3x² - x - 1 = 0. Solution: a=3, b=-1, c=-1 x = [-(-1) ± √((-1)² - 4(3)(-1))] / 2(3) x = [1 ± √(1 + 12)] / 6 x = [1 ± √13] / 6 Solutions: x = (1 + √13)/6 or x = (1 - √13)/6

Example 3: Using the Discriminant (High School)

Problem: How many real solutions does 4x² - 12x + 9 = 0 have? Solution: Calculate the discriminant (b² - 4ac). a=4, b=-12, c=9 Discriminant = (-12)² - 4(4)(9) = 144 - 144 = 0. Since the discriminant is 0, there is exactly one real solution.

Common Misconceptions

  1. Forgetting ±: When using the square root property or formula, include both positive and negative roots.
  2. Formula Errors: Incorrectly substituting values into the quadratic formula (especially signs).
  3. Factoring Errors: Incorrectly factoring the quadratic expression.
  4. Simplification Errors: Mistakes when simplifying radicals or fractions resulting from the formula.
  5. Ignoring a ≠ 0: The definition requires a to be non-zero; otherwise, it's a linear equation.

Applications in SAT

Quadratic equations are a major component of the SAT Math section ("Passport to Advanced Math"):

  1. Solving Equations: Using any of the methods (factoring, formula often most efficient).
  2. Interpreting Solutions: Understanding what the roots represent in context (e.g., time, distance).
  3. Analyzing Quadratic Functions: Finding the vertex, axis of symmetry, maximum/minimum values, intercepts of parabolas.
  4. Using the Discriminant: Determining the number/type of solutions without fully solving.
  5. Word Problems: Modeling real-world scenarios with quadratic equations (e.g., projectile motion).

Advanced Connections

Quadratic equations lead into:

  1. Advanced Functions - Analyzing parabolas and other polynomial functions.
  2. Complex Numbers - Arise when the discriminant is negative.
  3. Conic Sections: Parabolas are one type of conic section.
  4. Calculus: Finding maxima/minima of functions often involves solving derivatives set to zero, which can be quadratic.
  5. Physics: Modeling projectile motion and other physical phenomena.

Practice Problems

  1. Basic: Solve by factoring: x² - 8x + 15 = 0.
  2. Intermediate: Solve using the quadratic formula: 2x² + 4x - 3 = 0.
  3. Advanced: Find the value(s) of k such that x² + kx + 9 = 0 has exactly one real solution.
  4. SAT-Level: The height h, in feet, of an object thrown upwards is given by h(t) = -16t² + 64t + 80, where t is the time in seconds. After how many seconds does the object hit the ground (h = 0)?