Quadratic Equations Without Constant

A quadratic equation without a constant term is expressed in the form $ax^2 + bx = 0$. These equations are fundamental in mathematics with applications in diverse fields such as physics and economics. This guide assists in understanding, solving, and applying these equations effectively.

Key Concepts

Definitions

• Quadratic Equation: $ax^2 + bx = 0$.
• Zero Product Property: This property asserts that if $a \times b = 0$, then $a = 0$ or $b = 0$.

Solving Quadratic Equations Without a Constant Term

Characteristics

• Common Factor 'x'

  • These equations can be expressed as $x(ax+b)=0$.
  • Example: $2x^2 + 3x = 0$ simplifies to $x(2x + 3) = 0$.

• Graphical Features

  • These equations depict parabolas passing through the origin $(0,0)$.
  • The parabola is symmetric along the y-axis when $a=b$.

Process Steps

• Identify and Extract 'x'
  • Example: Convert $3x^2 + 6x = 0$ to $x(3x+6)=0$.
  • Utilise visual aids like algebra tiles for enhanced comprehension.

• Apply Zero Product Property
  • Set each factor to zero and solve individually.
  • For $x(3x + 6) = 0$: Solutions include $x = 0$ or $x = -2$.

Worked Examples

• Example 1: Solve $-4x^2 + 8x = 0$

  • First, factor out the common term: $-4x(x-2) = 0$
  • Using the zero product property:
    • Either $-4x = 0$, which gives $x = 0$
    • Or $x-2 = 0$, which gives $x = 2$
  • Therefore, the solutions are $x = 0$ and $x = 2$.

• Example 2: For $2x^2 + 4x = 0$

  • Factor out the common term $2x$: $2x(x + 2) = 0$
  • Using the zero product property:
    • Either $2x = 0$, which gives $x = 0$
    • Or $x + 2 = 0$, which gives $x = -2$
  • Therefore, the solutions are $x = 0$ and $x = -2$.

Graphing Quadratic Equations Without a Constant Term

Graph Properties

• Parabolic Shape: The vertex is located at the origin $(0,0)$.
• Intercepts: The graph intersects at the origin and has a second intercept at $x = -\frac{b}{a}$.

Techniques

• Manual Graphing: Mark intercepts and utilise symmetry for accuracy.
• Digital Tools: Various graphing tools enhance comprehension.

Real-World Applications

Scenarios and Implications

• Projectile Motion: Modelled by $h=vt-\frac{1}{2}gt^2$, it transforms to $t(v-\frac{g}{2}t)=0$ to forecast landing time.

• Economic Models: Consider a revenue model $100x-x^2$. Solving this optimises pricing strategies.

Translating Word Problems

• Recognise variables to set up equations.
• Convert scenarios into $ax^2 + bx = 0$ for resolution.

By mastering the concepts, techniques, and strategies for solving quadratic equations without a constant term, you enhance your academic and practical skills. Keep practising and apply these principles in real-world situations for enduring success.