Rational and Radical Equations

Understanding Extraneous Solutions

In this lesson:

• Solve rational equations using the LCD method
• Solve radical equations using isolate-raise-check
• Identify and explain extraneous solutions

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Solve rational equations using the LCD
2. Solve radical equations by isolating and raising to a power
3. Identify and discard extraneous solutions
4. Explain WHY extraneous solutions arise
5. State domain restrictions for both equation types

Recall: Fractions and the LCD

• LCD (least common denominator): the smallest expression divisible by every denominator
• Multiplying fractions: — multiply numerator only
• Denominator = 0: makes the fraction undefined — excluded from any domain

Can you find the LCD of and ?

A Puzzle: Where Did the Answer Go?

Look at this equation and follow the algebra step shown:

"Cancel from both sides" →

Now substitute back into the original:

Division by zero. The equation is broken.

Multiplying by Zero Breaks Everything

The step "cancel " is really multiplying both sides by

The multiplication property requires . When : we multiplied by .

An extraneous solution is a candidate that algebra produces but the original equation rejects.

Domain Restrictions Come Before Algebra

Before solving, identify every that makes a denominator zero — these are excluded.

Example: → domain: ,

Excluded values are automatically rejected — no algebra check needed.

Rational Equation: LCD Method in Action

Solve:

Domain: , . LCD:

Multiply both sides by :

Rational Equation: Solve and Check

From previous slide:

Check both: ✓ and ✓

Neither solution is 0 or −1, so both are valid.

Worked Example: Rational Equation (Extraneous)

Solve:

Domain: . LCD:

Multiply both sides by :

But violates the domain restriction .

Your Turn: Apply the LCD Method

Solve

1. What is the domain restriction?
2. What value does the algebra produce?
3. Is the solution valid or extraneous?

Try it before advancing.

Why Non-Reversible Steps Break the Chain

• Valid: multiply by a constant — the step is reversible (divide back)
• Invalid for some x: multiply by a variable expression — it equals zero for certain -values
• Multiplying by zero produces — true for any equation, meaning nothing

Checking Closes the Implication Chain

The check is a logical requirement — not arithmetic double-checking.

The implication: IF the equation holds AND , THEN

When , the assumption was false — the chain broke.

Checking in the original equation detects the break.

Predict: Can We Cancel Here?

Can we cancel from ?

A. Yes — canceling gives , false → no solution.

B. No — canceling is invalid because could be zero.

Commit to A or B, then advance.

Why Both Answers Reach the Same Place

Both A and B conclude: no solution — for different reasons.

• Canceling: is a contradiction → no solution ✓
• LCD path: multiply by → ; plus excluded ✓

Domain restrictions make explicit what canceling assumes silently.

Same Risk, Different Equation Type

Rational equations: multiplying by variable LCD might be multiplying by zero.

Radical equations use a similar risky step — which one?

• LCD step: valid only when the expression
• Squaring: valid forward, but introduces an extra branch on the return

The same checking logic applies to both.

Three Steps for Every Radical Equation

A radical equation has a variable inside a radical sign.

Always in this order:

1. Isolate the radical on one side
2. Raise both sides to the index power (square for √, cube for ∛)
3. Check all solutions in the original equation

Even-index radicals: domain requires radicand

Predict First: Isolate or Square?

A student sees and immediately squares both sides:

Check: . ✗

What did the student do wrong?

Why Squaring Before Isolating Fails

The error: . The correct expansion still contains a radical.

Correct approach: Isolate first: , then square: .

Check: ✓

Why Squaring Introduces Extraneous Solutions

Squaring loses sign information: means , so or

Radical Equation with No Extraneous Solution

Solve:

Domain: . Radical already isolated.

Square:

Check: ✓

Radical Equation: One Extraneous Solution

Solve:

Domain: . Square:

: ✓ : ✗

Cube Root: No Extraneous Solution Here

Solve:

Cube both sides:

Check: ✓

Cubing is one-to-one — no extraneous solutions possible.

Your Turn: Radical Equation with Check

Solve:

1. State the domain restriction
2. Square both sides (already isolated)
3. Solve the resulting equation
4. Check each candidate

Work through all four steps before advancing.

Checking: Required or Good Practice?

• Required: after any non-reversible step
• Always good practice: even when not required

Applying Three Steps to a Nested Radical

Solve:

Square outer radical:

Square inner radical:

Check: ✓

Who Kept a False Solution?

Check each student's answer in the original equation:

Student A: → reports and

Student B: → reports

Student C: → reports

Which student(s) retained an extraneous solution?

Mixed Practice: No Scaffolding Provided

For each: state domain, solve, check, give solution set.

Work alone before comparing with a partner.

What You Can Do Now

✓ Rational: multiply by LCD → solve → check

✓ Radical: isolate → raise to power → check

✓ Checking is logically required after non-reversible steps

Domain restrictions go first — before any algebra

Isolate the radical before squaring

Odd-index roots don't produce extraneous solutions

Next Lesson: Solving Quadratic Equations

HSA.REI.B.4 — the quadratic formula always produces two candidates.

The same discipline applies: interpret both in context and decide which are valid.

The checking habit you built today carries directly forward.