Session 02-04 - Fractional, Radical & Cubic Equations

Section 02: Equations & Problem-Solving Strategies

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

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 Presentations

Solutions from Tasks 02-03

20 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!

Key Concept Review

Building on What You Know

Connecting to today’s content

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
  • Advanced factoring for cubics

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: \(x \neq 3\)

Always 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!

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: \(\sqrt{9} + \sqrt{4} = 3 + 2 = 5\) ✓

Cubic Equations

Factoring Strategies

Building on what you know

You already know these from Section 01:

  • Sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
  • Difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

Example: \(x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)\)

For simple cubics, try small integer values:

  • Test \(x = 0, \pm 1, \pm 2, \pm 3\)…
  • If \(x = a\) works, then \((x - a)\) is a factor
  • Use long division to find the other factor

For \(x^3 + px^2 + qx + r\):

  • Look for common factors first
  • Try grouping in pairs
  • Example: \(x^3 + 2x^2 - x - 2 = x^2(x + 2) - 1(x + 2) = (x + 2)(x^2 - 1)\)

Solving Cubic Equations

Step-by-step approach

Solve: \(x^3 - 6x^2 + 11x - 6 = 0\)

  • Try simple values: Test \(x = 1\): \(1 - 6 + 11 - 6 = 0\) ✓
  • So \((x - 1)\) is a factor! Now we need to find what it multiplies with
  • Divide: We know \(x^3 - 6x^2 + 11x - 6 = (x - 1) \times ?\)
  • By inspection or long division: \((x - 1)(x^2 - 5x + 6) = 0\)
  • Factor the quadratic: \((x - 1)(x - 2)(x - 3) = 0\)
  • Solutions: \(x = 1, 2, 3\)

Special Case: Difference/Sum of Cubes

Using what you already know

Solve \(x^3 - 27 = 0\)

  • Recognize as difference of cubes: \(x^3 - 3^3 = 0\)
  • Factor: \((x - 3)(x^2 + 3x + 9) = 0\)
  • From first factor: \(x = 3\)
  • From second factor: Use quadratic formula
  • \(x = \frac{-3 \pm \sqrt{9 - 36}}{2} = \frac{-3 \pm \sqrt{-27}}{2}\) (no real solutions)
  • Only real solution: \(x = 3\)

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. Factor and solve: \(x^3 - 27 = 0\)

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

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

  6. Solve: \(x^3 + 2x^2 - 5x - 6 = 0\)

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!

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

Master these essential concepts

  • Domain restrictions must be checked FIRST in rational equations
  • LCD method clears fractions efficiently
  • Squaring introduces extraneous solutions - always verify!
  • Cubic factoring uses rational root theorem and special forms
  • Cross multiplication works for proportion equations
  • Real-world rates often involve rational equations

Final Assignment

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. Factor: \(x^3 + 8\)

Next Session Preview

Session 02-05: Exponential, Logarithmic & Complex Word Problems

  • Exponential growth and decay models
  • Logarithmic equations and applications
  • Complex multi-step word problems
  • Financial mathematics applications

Review logarithm properties from Section 01 - we’ll use them extensively!

Homework Assignment

Complete Tasks 02-04

Focus on:

  • Domain restriction practice
  • Checking all solutions in radical equations
  • Factoring cubic expressions
  • Real-world rate problems

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