Session 02-04 - Fractional, Radical & Cubic Equations

Section 02: Equations & Problem-Solving Strategies

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

Quick Review of Previous Methods

10 minutes - individual work, then peer review

Solve by factoring: \(x^2 - 9x + 20 = 0\)
Use the quadratic formula: \(2x^2 + 3x - 1 = 0\)
Solve the biquadratic: \(x^4 - 5x^2 + 4 = 0\)
Find the discriminant of: \(x^2 - 6x + 9 = 0\)
Factor completely: \(x^3 - 8\)

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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

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

Find domain restrictions: \(x \neq 0, x \neq 1\)
Find LCD: \(x(x-1)\)
Multiply all terms by LCD: \(2(x-1) + 3x = x(x-1)\)
Expand: \(2x - 2 + 3x = x^2 - x\)
Rearrange: \(x^2 - 6x + 2 = 0\)
Solve: \(x = \frac{6 \pm \sqrt{36-8}}{2} = 3 \pm \sqrt{7}\)
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

Solve and state domain: \(\frac{3}{x-2} + \frac{1}{x} = 1\)
Solve: \(\sqrt{2x + 1} = x - 2\)
Factor and solve: \(x^3 - 27 = 0\)
Solve: \(\frac{x}{x+1} = \frac{2}{x-1}\)
Solve: \(\sqrt{x + 7} - \sqrt{x} = 1\)
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.

Find the time when this relationship holds for \(C = 6\)
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:

\(\frac{2}{x+1} - \frac{1}{x-1} = \frac{1}{2}\)
\(\sqrt{3x - 2} = x\)
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

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