Thirty-Six

Unit 4 · Expressions, Quadratics & PolynomialsA6 · Rational & Radical · Structures

Rational & Radical Equations

Respect domains, clear denominators, isolate radicals, and reject extraneous roots.

  • 30 min
  • Unlocks: Cryogenic Insulation
  • Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

extraneous solution
a candidate created by an irreversible algebraic step

Step 2 of 7Try it firstDone

Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

Step 3 of 7LearnDone

The big idea

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The decision that matters

Write excluded denominator values first and check radical candidates in the original. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Write excluded denominator values first and check radical candidates in the original. Keep exact values through the setup; round only when the stem asks.
Core relationship
A=B⟹B≥0\sqrt{A}=B\quad\Longrightarrow\quad B\ge0

Squaring can create candidates that fail the original equation, so substitute every candidate back into the unsquared equation.

Algebra or Desmos?

Algebra produces candidate solutions; substitution into the original equation decides which candidates survive. Desmos can show intersections for a radical equation, but excluded denominator values and domain conditions still require reasoning.

Explore in Desmos

Check radical-equation candidates

Graph the equation, record both algebraic candidates, and check each in the original.

The calculator is live: change anything and see what happens.

Check radical-equation candidates

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Trap

The main trap

Cancel factors only; terms joined by addition cannot cancel.

Step 4 of 7Worked examplesDone

Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

Solve x+3x−2=2\frac{x+3}{x-2}=2, with x≠2x\ne2.
Exclude x=2x=2.

Which equation results from multiplying both sides by x−2x-2?

The candidate is x=7x=7.

Why is it valid?

Worked example

Example 2: ACT twist

Problem

Solve x+6=x\sqrt{x+6}=x and reject any extraneous candidate.
The right side forces x≥0x\ge0. Square: x+6=x2x+6=x^2.

Which candidate must be rejected after checking the original equation?

Only x=3x=3 works in the original.

Why does x=−2x=-2 fail?

Faded example · you finish it

Example 3: You finish it

Problem

Solve 2x−1=5\sqrt{2x-1}=5.
2x−1=252x-1=25

Solve 2x−1=252x-1=25, then check the original radical equation.

Desmos way

Intersection check for the radical equation

Graph the equation, record both algebraic candidates, and check each in the original.

Intersection check for the radical equation

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Step 5 of 7PracticeDone

Practice

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Step 6 of 7Spot the mistakeDone

Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

Step 7 of 7Exit ticketDone

Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.