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Radical and Rational Equations

  • Grade 11
  • Mathematics
  • A-REI.A.2
  • Algebra
  • Free lesson

Aligned to A-REI.A.2 — Common Core State Standards for Mathematics.

What this lesson teaches

Solve a radical equation by isolating the root and squaring both sides. Squaring can introduce extraneous solutions that do not satisfy the original equation, so you must check every answer.

Worked examples

Example 1

Solve √(x + 5) = x − 1. Square both sides: x + 5 = x^2 − 2x + 1, so x^2 − 3x − 4 = 0, giving (x − 4)(x + 1) = 0 and x = 4 or x = −1. Check: x = 4 gives √9 = 3 ✓; x = −1 gives √4 = 2 but x − 1 = −2 ✗, so x = 4 only.

Example 2

Solve 3/(x − 2) + 1 = 5/(x − 2). Multiply every term by (x − 2): 3 + (x − 2) = 5, so x + 1 = 5 and x = 4. Check that x = 4 does not make the denominator zero — it does not, since 4 − 2 = 2. The answer is x = 4.

Example 3

Common mistake: skipping the check. Squaring and clearing denominators can CREATE solutions that do not satisfy the original. Solve 1/(x − 3) = 4/(x² − 9): multiplying by (x + 3)(x − 3) gives x + 3 = 4, so x = 1 — check it, and it works. But the same method on 1/(x − 3) = 6/(x² − 9) gives x = 3, which makes both denominators zero. Every radical or rational equation needs its answers substituted back.

Practice questions

  1. Solve √(2x + 1) = 3.
  2. Solve √(x + 6) = x and check for extraneous solutions.
  3. Solve 1/(x − 2) = 3 for x.

Watch the lesson

Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.

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En español

Ecuaciones Radicales y Racionales

Resuelve una ecuación radical aislando la raíz y elevando ambos lados al cuadrado. Elevar al cuadrado puede introducir soluciones extrañas que no satisfacen la ecuación original, por lo que debes verificar cada respuesta.

Ejemplo 1: Resuelve √(x + 5) = x − 1. Eleva al cuadrado ambos lados: x + 5 = x^2 − 2x + 1, así que x^2 − 3x − 4 = 0, dando (x − 4)(x + 1) = 0 y x = 4 o x = −1. Verifica: x = 4 da √9 = 3 ✓; x = −1 da √4 = 2 pero x − 1 = −2 ✗, así que solo x = 4.

Ejemplo 2: Resuelve 3/(x − 2) + 1 = 5/(x − 2). Multiplica cada término por (x − 2): 3 + (x − 2) = 5, así que x + 1 = 5 y x = 4. Comprueba que x = 4 no anule el denominador — no lo hace, ya que 4 − 2 = 2. La respuesta es x = 4.

Ejemplo 3: Error común: saltarse la comprobación. Elevar al cuadrado y quitar denominadores puede CREAR soluciones que no cumplen la ecuación original. Resuelve 1/(x − 3) = 4/(x² − 9): multiplicar por (x + 3)(x − 3) da x + 3 = 4, así que x = 1 — compruébalo y sí funciona. Pero el mismo método en 1/(x − 3) = 6/(x² − 9) da x = 3, que anula ambos denominadores. Toda ecuación radical o racional necesita que sus respuestas se sustituyan de vuelta.

More Algebra lessons

Zeros and Factors of PolynomialsA-APR.B.3Rewriting Expressions (Completing the Square)A-SSE.B.3Sum of a Finite Geometric SeriesA-SSE.B.4The Remainder TheoremA-APR.B.2Proving Polynomial IdentitiesA-APR.C.4Rewriting Rational ExpressionsA-APR.D.6

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