IB Mathematics AA SL Solving equations Study Notes

IB Mathematics AA SL Solving equations Study Notes

IB Mathematics AA SL Solving equations Study Notes Offer a clear explanation of Solving equations , including various formula, rules, exam style questions as example to explain the topics. Worked Out  examples and common problem types provided here will be sufficient to cover for topic Solving equations

Solving Equations Graphically and Analytically

Solving Equations Graphically and Analytically

There are two common approaches to solving equations involving exponential and logarithmic terms:

  • Graphically: Plot the functions on a graphing calculator or software (e.g., Desmos, GeoGebra, GDC) and find the x-values where the curves intersect (the solutions).
  • Analytically: Use algebraic techniques such as substitution, factoring, logarithms, or applying the quadratic formula to solve the equation exactly.

For exponential equations, substitution often reduces the equation to a quadratic form, making it easier to solve.

Example: Solve the equation analytically and graphically:

$ e^{2x} – 5e^x + 4 = 0 $

▶️Answer/Explanation

Let \( y = e^x \). Then the equation becomes:

$ y^2 – 5y + 4 = 0 $

Solve the quadratic

Factor the quadratic:

$ (y – 4)(y – 1) = 0 $

So, \( y = 4 \) or \( y = 1 \).

Back-substitute \( y = e^x \)

  • \( e^x = 4 \Rightarrow x = \ln 4 \)
  • \( e^x = 1 \Rightarrow x = \ln 1 = 0 \)

$ x = 0 \quad \text{or} \quad x = \ln 4 $

Graphical solution

Plot the function \( f(x) = e^{2x} – 5e^x + 4 \) using technology. The x-values where the graph crosses the x-axis correspond to the solutions:
The graph crosses at \( x = 0 \) and \( x = \ln 4 \approx 1.386 \).

Use of Technology to Solve Equations

Use of Technology to Solve Equations

For some equations, an analytic (exact algebraic) solution may be difficult or impossible. In these cases, we use graphing technology (like TI-84, Desmos, or GeoGebra) to find approximate solutions by locating intersections or zeros.

General steps:

  1. Enter the left-hand side and right-hand side as two separate functions into your graphing tool.
  2. View the graph and adjust window settings if necessary to see intersections.
  3. Use the “intersect” or “zero” function to find solutions.
  4. Record the approximate x-values where the graphs intersect (or where the function crosses the x-axis).

Example : Solve \( e^x = \sin x \) using technology.

▶️Answer/Explanation
  1. Graph \( f(x) = e^x \).
  2. Graph \( g(x) = \sin x \).
  3. Look for intersections (they may occur where \( \sin x \) is positive).
  4. Use the “intersection” tool to find:
    • \( x \approx 0 \) (since \( e^0 = 1 \) and \( \sin 0 = 0 \); near 0, no solution, but check close to 0)
    • Solution occurs approximately at \( x \approx -3.188 \)

The solution is approximately:

\( x \approx -3.188 \)

Example : Solve \( x^4 + 5x – 6 = 0 \) using technology.

▶️Answer/Explanation
  1. Graph \( f(x) = x^4 + 5x – 6 \).
  2. Use the “zero” or “root” tool to find where the graph crosses the x-axis.
  3. The calculator shows:
    • \( x \approx -2 \)
    • \( x \approx 1 \)

Applications of Graphing Skills and Solving Equations in Real Life

Applications of Graphing Skills and Solving Equations in Real Life

Graphing skills and solving equations are essential in modeling and analyzing many real-life contexts. Examples include:

  • Predicting profit or cost in business models
  • Analyzing the height of a projectile over time in physics
  • Determining when supply equals demand in economics
  • Estimating population growth with exponential functions
  • Modeling cooling or heating rates with exponential decay or growth

By graphing functions or solving equations, we can:

  • Find when certain thresholds are reached (e.g., break-even point)
  • Identify maximum or minimum values (e.g., highest point of a rocket)
  • Determine when two quantities are equal (e.g., balance point between income and expense)

Example: A company models its profit (in $) as a function of the number of units sold by:

$ P(x) = -2x^2 + 200x – 5000 $

Where \( x \) is the number of units produced and sold. Find how many units must be sold to maximize profit, and what that maximum profit is.

▶️Answer/Explanation

Graph \( P(x) = -2x^2 + 200x – 5000 \) using technology (e.g., Desmos, GeoGebra).

the vertex, as this quadratic opens downward (maximum point).

The vertex occurs at: $ x = -\frac{b}{2a} = -\frac{200}{2(-2)} = 50 $

Substitute: $ P(50) = -2(50)^2 + 200(50) – 5000 = -5000 + 10000 – 5000 = 0 $

The company breaks even at 50 units (no profit, no loss).

Using graphing tools helps to confirm this visually and find approximate solutions if coefficients were more complex.

Scroll to Top