Solving equations (algebraic/graphical) - Mathematics: Analysis & Approaches IB Study Notes

Solving equations involves finding values of the variable that satisfy an equation, making both sides equal. In IB Mathematics: Analysis & Approaches, you'll encounter two primary methods:

Algebraic Methods manipulate equations using mathematical operations to isolate the variable. Key techniques include:

• Linear equations: ax + b = c, solved by inverse operations
• Quadratic equations: ax² + bx + c = 0, solved using factorization, completing the square, or the quadratic formula: x = (-b ± √(b² - 4ac))/(2a)
• Rational equations: equations with fractions, solved by finding common denominators
• Exponential equations: equations with variables in exponents (e.g., 2^x = 8)
• Logarithmic equations: using log properties to solve for variables

Graphical Methods involve finding intersection points of graphs. When solving f(x) = g(x), plot both y = f(x) and y = g(x); solutions are the x-coordinates where graphs intersect.

Key Principle: An equation with n solutions has n intersection points graphically

Important terminology:

• Root/Zero: value making f(x) = 0
• Solution: value satisfying the equation
• Exact solution: algebraic form (e.g., √3, ln 5)
• Approximate solution: decimal form (e.g., 1.732, 1.609)

The discriminant (Δ = b² - 4ac) determines quadratic solution types:

• Δ > 0: two real solutions
• Δ = 0: one repeated solution
• Δ < 0: no real solutions (two complex solutions)

Mnemonic: "Algebra Answers Accurately" — Always check solutions by substituting back into original equations.

Solving equations underpins countless real-world applications across science, economics, and engineering.

Break-Even Analysis in Business: Companies solve cost equations to find profitability thresholds. If production costs are C(x) = 5000 + 20x and revenue is R(x) = 50x, solving 5000 + 20x = 50x gives the break-even quantity (x = 166.67 units). Graphically, this is where cost and revenue lines intersect.

Projectile Motion in Physics: When launching a rocket, its height h(t) = -4.9t² + 50t + 100 follows a parabolic path. Finding when it hits ground means solving -4.9t² + 50t + 100 = 0. The quadratic formula reveals landing time, while graphing shows the complete trajectory.

Pharmacokinetics: Drug concentration in bloodstream follows exponential decay: C(t) = C₀e^(-kt). Doctors solve this equation to determine when concentration falls to safe levels for next dosage. For example, solving 100e^(-0.3t) = 25 finds when concentration drops to 25% of original.

Analogy: The Detective Approach Think of solving equations like detective work. Algebraic methods are interrogating suspects with logic and evidence, systematically eliminating possibilities until identifying the culprit (solution). Graphical methods are examining security footage (graphs) to see where suspects' paths cross (intersections). Sometimes one method is faster; often using both confirms your answer.

Real-world insight: Engineers rarely solve equations by hand—software does it. However, understanding which method suits each situation and interpreting solutions remains essential human expertise.

The graphical approach particularly helps visualize number of solutions and approximate values when algebraic methods become complex.