Quadratic Formula Core Concepts

Understanding quadratic equations is essential in mathematics. Quadratic equations take the form $ax^2 + bx + c = 0$, where:

• Coefficient $a$, $b$, $c$: These are the numbers multiplying the variables in equations, serving as factors for terms.
• Quadratic Equation: A polynomial equation of degree two.

Key Techniques

Quadratic equations can be solved using various methods:

• Quadratic Formula: Provides a universal approach to solving quadratic equations.
• Completing the Square: Converts quadratics into a perfect square trinomial.
• Factoring: Simplifies equations directly when factor pairs matching the terms are identifiable.

Introduction to Quadratic Equations

Quadratic equations are commonly encountered in various real-world contexts and mathematical applications.

• Physics/Engineering: Used in analysing projectiles' trajectories.
• Economics: Utilised in models to aid in profit maximisation.
• Everyday Scenarios: Applied in calculating garden areas.

Properties of Parabolas

Parabolas, which are the graphical representations of quadratic equations, possess key features:

• Vertex: The highest or lowest point on the parabola.
• Axis of Symmetry: The vertical line passing through the vertex.
• X-intercepts: Points where the parabola intersects the x-axis.

Understanding the Discriminant

The discriminant is a component of the quadratic formula:

• Discriminant $b^2 - 4ac$:
  • Positive: Indicates two real, distinct roots.
  • Zero: Indicates one real repeated root.
  • Negative: Indicates two complex roots.

Problem Set A: Quadratic Formula

Basic Problem

Solve $2x^2 + 3x - 2 = 0$.

• Worked Example:
  • Identify the coefficients: $a=2$, $b=3$, $c=-2$.
  • Apply the quadratic formula: $x = \frac{-3 \pm \sqrt{3^2 - 4 \times 2 \times (-2)}}{2 \times 2}$ $x = \frac{-3 \pm \sqrt{9 + 16}}{4}$ $x = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}$
  • This gives us: $x_1 = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}$ $x_2 = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$
  • Solutions: $x_1 = \frac{1}{2}$, $x_2 = -2$.

Intermediate Problem

Solve $x^2 - 4x + 5 = 0$ using the discriminant to interpret roots.

• Worked Example:
  • Calculate the discriminant: $b^2 - 4ac = (-4)^2 - 4 \times 1 \times 5 = 16 - 20 = -4$
  • Since the discriminant is negative, the equation has two complex roots.
  • For completeness, the roots are: $x = \frac{4 \pm \sqrt{-4}}{2} = 2 \pm i$
  • Solutions: $x_1 = 2 + i$, $x_2 = 2 - i$

Problem Set B: Completing the Square

Basic Example

Complete the square for: $x^2 + 6x + 8 = 0$.

• Worked Example:
  • Rearrange to standard form: $x^2 + 6x + 8 = 0$ becomes $x^2 + 6x = -8$
  • Half the coefficient of $x$: $\frac{6}{2} = 3$
  • Square this value: $3^2 = 9$
  • Add and subtract this value: $x^2 + 6x + 9 - 9 = -8$
  • Factor the perfect square: $(x+3)^2 - 9 = -8$
  • Rearrange: $(x+3)^2 = 1$
  • Take the square root: $x + 3 = \pm 1$
  • Solve for $x$: $x = -3 \pm 1$
  • Solutions: $x_1 = -2$, $x_2 = -4$

Worked Examples and Exam Tips

• Verify Solutions:
  • Ensure accuracy by substituting solutions back into the original equation.
  • Graph solutions to verify roots and examine the parabola's shape.

Visualising Solutions

Using graphing tools enhances comprehension of solution processes and parabola characteristics.

• Emphasising the vertex, symmetry, and x-intercepts ensures precise verification of solutions.

Checking Solutions

Verification Methods:

• Substitute roots back into the original equation.
• Cross-check solutions graphically.
• Use discriminant sign analysis to confirm root types (real/imaginary).

Mastering the quadratic formula, completing the square, and graphical analysis provides a robust foundation for effectively solving quadratic equations.