Session 02-03 - Fractional, Radical, Exponential & Logarithmic Equations

Section 02: Equations & Problem-Solving Strategies

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Quick Review of Previous Methods

10 minutes - individual work, then peer review

  1. Solve by factoring: \(x^2 - 9x + 20 = 0\)

  2. Use the quadratic formula: \(2x^2 + 3x - 1 = 0\)

  3. Solve the biquadratic: \(x^4 - 5x^2 + 4 = 0\)

  4. Find the discriminant of: \(x^2 - 6x + 9 = 0\)

  5. Factor completely: \(x^3 - 8\)

. . .

Check your factoring skills - they’re crucial for today!

Homework Discussion - 15 Minutes

Solution from Tasks 02-02

15 minutes - presentation and discussion

  • Share your break-even analysis solutions
  • Discuss method selection strategies
  • Present any challenging biquadratic equations
  • Review work rate problems if time permits

. . .

Today we extend our toolkit to handle more complex equation types!

Learning Objectives

Learning Objectives

By the end of this session, you can:

  • Identify domain restrictions in fractional equations before solving
  • Solve radical equations and detect extraneous solutions
  • Solve advanced exponential equations via substitution and logarithms
  • Solve logarithmic equations and systems, checking domains

Key Concept Review

Building on What You Know

Your current equation-solving toolkit:

  • Zero Product Property: If \(AB = 0\), then \(A = 0\) or \(B = 0\)
  • Quadratic methods: Factoring, formula, completing square
  • Substitution: Transform complex equations to simpler ones

. . .

Today’s additions:

  • Domain restrictions for rational equations
  • Checking for extraneous solutions
  • Solving techniques for exponential and logarithmic equations

Fractional Equations

Understanding Domain Restrictions

Critical new concept

Domain: The set of all valid input values

For fractional equations, we must exclude values that make denominators zero!

Example: \(\frac{1}{x - 3}\) has domain restriction: \(x \neq 3\)

. . .

WarningAlways Check First!

Before solving any fractional equation, identify ALL domain restrictions.

Solving Fractional Equations

Method: Clear denominators by multiplying by LCD

Example: Solve \(\frac{2}{x} + \frac{3}{x-1} = 1\)

  1. Find domain restrictions: \(x \neq 0, x \neq 1\)
  2. Find LCD: \(x(x-1)\)
  3. Multiply all terms by LCD: \(2(x-1) + 3x = x(x-1)\)
  4. Expand: \(2x - 2 + 3x = x^2 - x\)
  5. Rearrange: \(x^2 - 6x + 2 = 0\)
  6. Solve: \(x = \frac{6 \pm \sqrt{36-8}}{2} = 3 \pm \sqrt{7}\)
  7. Check domain: Both solutions valid! ✓

Common Fractional Patterns

Recognize these structures

\(\frac{a}{x} + \frac{b}{x} = c\)

  • Combine: \(\frac{a+b}{x} = c\)
  • Solve: \(x = \frac{a+b}{c}\)

\(\frac{a}{b} = \frac{c}{d}\)

  • Cross multiply: \(ad = bc\)
  • Much faster than finding LCD!

\(\frac{\frac{a}{x}}{\frac{b}{y}} = c\)

  • Simplify: \(\frac{ay}{bx} = c\)
  • Then solve normally

Application: Work Rate Problem I

Practical example with domain restrictions

Two teams can complete a project. Their combined work rate equation is: \[\frac{1}{x} + \frac{1}{x+5} = \frac{1}{3}\] where \(x\) is the time (in days) for Team A to complete the project alone.

  • Domain restrictions: \(x > 0\) (time must be positive)
  • Also \(x \neq -5\), but this is already excluded by \(x > 0\)
  • Solving: Find LCD = \(3x(x+5)\)

Application: Work Rate Problem II

  • Multiply through: \(3(x+5) + 3x = x(x+5)\)
  • Expand: \(3x + 15 + 3x = x^2 + 5x\)
  • Rearrange: \(x^2 - x - 15 = 0\)
  • Using quadratic formula: \(x = \frac{1 \pm \sqrt{1 + 60}}{2} = \frac{1 \pm \sqrt{61}}{2}\)
  • Solutions: \(x \approx 4.41\) or \(x \approx -3.41\)
  • Check domain: Only \(x = 4.41\) days is valid (positive time)
  • Business meaning: Team A takes 4.41 days alone, Team B takes 9.41 days alone
  • Together they complete it in 3 days as required

Break - 10 Minutes

Radical Equations

Solving Strategy

Isolate, square, check!

Key principle: To eliminate a square root, square both sides

Critical warning: Squaring can introduce extraneous solutions!

Example: \(\sqrt{x + 3} = x - 1\)

  • Square both sides: \(x + 3 = (x - 1)^2\)
  • Expand: \(x + 3 = x^2 - 2x + 1\)
  • Rearrange: \(x^2 - 3x - 2 = 0\)
  • Solutions: \(x = \frac{3 \pm \sqrt{9 + 8}}{2} = \frac{3 \pm \sqrt{17}}{2}\)
  • \(x_1 = \frac{3 + \sqrt{17}}{2} \approx 3.56\), \(x_2 = \frac{3 - \sqrt{17}}{2} \approx -0.56\)

Checking for Extraneous Solutions

Essential verification step

Check \(x_1 \approx 3.56\):

  • Left: \(\sqrt{3.56 + 3} = \sqrt{6.56} \approx 2.56\)
  • Right: \(3.56 - 1 = 2.56\) ✓

Check \(x_2 \approx -0.56\):

  • Left: \(\sqrt{-0.56 + 3} = \sqrt{2.44} \approx 1.56\)
  • Right: \(-0.56 - 1 = -1.56\) ✗

. . .

Only \(x_1\) is valid! Always check radical equation solutions!

Seeing the Extraneous Solution

. . .

Squaring both sides merges \(y = x - 1\) and \(y = -(x-1)\) into one equation - the second candidate solves the mirrored line, not the original!

Multiple Radicals

More complex scenarios

Solve: \(\sqrt{x + 5} + \sqrt{x} = 5\)

  • Isolate one radical: \(\sqrt{x + 5} = 5 - \sqrt{x}\)
  • Square: \(x + 5 = 25 - 10\sqrt{x} + x\)
  • Simplify: \(5 = 25 - 10\sqrt{x}\)
  • Isolate: \(10\sqrt{x} = 20\), so \(\sqrt{x} = 2\)
  • Solution: \(x = 4\)
  • Check: \(\sqrt{9} + \sqrt{4} = 3 + 2 = 5\) ✓

Calculator: The SOLVE Function I

Newton’s method for solving any equation:

SOLVE (accessed via SHIFT + CALC) uses Newton’s approximation method to find solutions.

. . .

  1. Enter an equation (e.g., x² + B = 0)
  2. Press SHIFT + CALC (SOLVE)
  3. Enter an initial guess for x and values for other variables
  4. Press = to find the solution

Calculator: The SOLVE Function II

Newton’s method for solving any equation:

Example: Solve \(x^2 - 2 = 0\) (find \(\sqrt{2}\))

  • Enter: ALPHA x² - 2 → SHIFT CALC → initial guess: 1 =
  • Result: \(x = 1.414213562\) (which is \(\sqrt{2}\))

. . .

SOLVE uses numerical methods, so:

  • Results depend on your initial guess
  • Multiple solutions require different starting points
  • The closer your guess, the faster and more reliable the result

Advanced Exponential Equations

Equations with Different Bases

When you can’t make bases equal

Solve: \(3^x \cdot 5^{x-1} = 45\)

  • Rewrite: \(3^x \cdot \frac{5^x}{5} = 45\)
  • Simplify: \(\frac{3^x \cdot 5^x}{5} = 45\)
  • Combine: \(\frac{(3 \cdot 5)^x}{5} = 45\)
  • So: \(15^x = 225 = 15^2\)
  • Therefore: \(x = 2\)

. . .

Look for ways to combine or separate bases strategically!

Mixed Exponential Systems

Solving simultaneous exponential equations

\(2^x + 2^y = 12\)

\(2^x - 2^y = 4\)

  • Let \(u = 2^x\) and \(v = 2^y\) for simplicity
  • System becomes: \(u + v = 12\) and \(u - v = 4\)
  • Add equations: \(2u = 16\), so \(u = 8\)
  • Subtract: \(2v = 8\), so \(v = 4\)
  • Therefore: \(2^x = 8 = 2^3\), giving \(x = 3\)
  • And: \(2^y = 4 = 2^2\), giving \(y = 2\)

Exponential Inequalities

New territory: solving inequalities

Solve: \(2^{x+1} > 8^{x-1}\)

  • Rewrite right side: \(8^{x-1} = (2^3)^{x-1} = 2^{3(x-1)}\)
  • Inequality becomes: \(2^{x+1} > 2^{3x-3}\)
  • Since base 2 > 1, we can compare exponents directly
  • \(x + 1 > 3x - 3\)
  • \(4 > 2x\)
  • Solution: \(x < 2\)

Advanced Logarithmic Equations

Equations with Mixed Bases

Using change of base strategically

Solve: \(\log_2(x) \cdot \log_x(8) = 3\)

  • Use change of base: \(\log_x(8) = \frac{\log_2(8)}{\log_2(x)} = \frac{3}{\log_2(x)}\)
  • Substitute: \(\log_2(x) \cdot \frac{3}{\log_2(x)} = 3\)
  • Simplify: \(3 = 3\) ✓
  • This is always true for any valid \(x\)!
  • Domain restriction: \(x > 0, x \neq 1\)
  • Solution: \(x \in (0,1) \cup (1,\infty)\)

Logarithmic Systems

Multiple equations with logs

\(\log(x) + \log(y) = 2\)

\(\log(x) - \log(y) = 1\)

  • Add equations: \(2\log(x) = 3\), so \(\log(x) = 1.5\)
  • Therefore: \(x = 10^{1.5} = 10\sqrt{10}\)
  • Subtract second from first: \(2\log(y) = 1\)
  • So: \(\log(y) = 0.5\), thus \(y = \sqrt{10}\)
  • Check: \(\log(10\sqrt{10}) + \log(\sqrt{10}) = 1.5 + 0.5 = 2\) ✓

Think-Pair-Share: Which Method When?

2 minutes alone, 3 minutes with your neighbour

Match each equation type to the best first move:

  1. \(5^{x+2} = 125\) · b) \(3^{2x} - 10 \cdot 3^x + 9 = 0\) · c) \(\log(x) + \log(x-3) = 1\) · d) \(2^x = 7\)
  • Same base possible? → rewrite and compare exponents
  • Quadratic pattern? → substitute \(u = a^x\)
  • Sum of logs? → combine with the product rule, then convert
  • No common base? → take logarithms of both sides

Guided Practice

Individual Exercises

Work independently

  1. Solve and state domain: \(\frac{3}{x-2} + \frac{1}{x} = 1\)

  2. Solve: \(\sqrt{2x + 1} = x - 2\)

  3. Solve: \(3^{2x} - 4 \cdot 3^x + 3 = 0\)

  4. Solve: \(\frac{x}{x+1} = \frac{2}{x-1}\)

  5. Solve: \(\sqrt{x + 7} - \sqrt{x} = 1\)

  6. Solve: \(\log_2(x) + \log_2(x - 2) = 3\)

Coffee Break - 15 Minutes

Application & Extension

Production Rate Problem

Find the individual times.

Machine A can complete an order in \(x\) hours. Machine B takes 3 hours longer. Working together, they complete it in 2 hours.

  • A’s rate: \(\frac{1}{x}\) orders/hour, B’s rate: \(\frac{1}{x+3}\) orders/hour
  • Combined: \(\frac{1}{x} + \frac{1}{x+3} = \frac{1}{2}\)
  • Solve: \(2(x+3) + 2x = x(x+3)\)
  • \(x^2 - x - 6 = 0\)
  • \((x - 3)(x + 2) = 0\)
  • Since \(x > 0\): Machine A takes 3 hours, B takes 6 hours

Investment Growth

Compound interest with radicals

An investment grows according to: \(A = P\sqrt{1 + 0.2t}\)

If €1,000 grows to €1,500, find the time \(t\).

  • Set up: \(1500 = 1000\sqrt{1 + 0.2t}\)
  • Simplify: \(\sqrt{1 + 0.2t} = 1.5\)
  • Square: \(1 + 0.2t = 2.25\)
  • Solve: \(0.2t = 1.25\)
  • Time: \(t = 6.25\) years

. . .

Do we have to check the domain restrictions?

Collaborative Problem-Solving

Complex Rate Challenge

Work in pairs to solve this problem

A chemical reaction follows the rate equation: \[\frac{C}{t} + \frac{C}{t+2} = 3\]

where \(C\) is concentration and \(t\) is time in hours.

  1. Find the time when this relationship holds for \(C = 6\)
  2. Verify your solution makes physical sense

. . .

Consider domain restrictions and check all solutions!

Epidemic Modeling Challenge

A disease spreads through a population of 10,000. The infected count follows: \[I = \frac{10000}{1 + 99e^{-0.5t}}\]

  1. How many are initially infected?
  2. When will half the population be infected?
  3. If a vaccine reduces the spread rate by 40%, modify the model

This is a logistic growth model - different from pure exponential!

Spot the Error

Find and fix the mistake

A student solved \(\sqrt{x + 4} = x - 2\):

Square both sides with \(x + 4 = x - 2\), therefore: \(4 = -2\), which is impossible. So there’s no solution.

What went wrong?

Wrap-Up & Key Takeaways

Key Takeaways

Master these essential concepts

  • Domain restrictions must be checked FIRST in rational equations
  • LCD method clears fractions efficiently
  • Squaring introduces extraneous solutions - always verify!
  • Exponential equations: substitute \(u = a^x\) or take logs strategically
  • Logarithmic equations: combine logs, convert, and check domains
  • Cross multiplication works for proportion equations
  • Real-world rates often involve rational equations

Final Assessment

10 minutes - individual assessment

Solve the following:

  1. \(\frac{2}{x+1} - \frac{1}{x-1} = \frac{1}{2}\)

  2. \(\sqrt{3x - 2} = x\)

  3. Solve: \(4^x - 3 \cdot 2^x = -2\)

Next Session: Mock Exam 02

Session 02-04 is exam only - no lecture

  • 90 minutes working time plus 15 minutes reading time, 50 points
  • Covers Sections 01 and 02: algebra, systems of equations, exponential growth and word problems
  • Allowed: non-programmable calculator, drawing instruments, monolingual dictionary
  • Not allowed: notes, formula sheets, any electronic devices
  • Counts toward the 60% of test points needed for exam admission

. . .

Review ALL equation-solving methods from Sections 01 and 02 - linear, quadratic, cubic, fractional, radical, exponential, and logarithmic!

Homework Assignment

Complete Tasks 02-03!

Focus on:

  • Domain restriction practice
  • Checking all solutions in radical equations
  • Exponential and logarithmic equations
  • Real-world rate problems

. . .

Remember: In fractional equations, always identify restrictions before solving!