Rational and Radical Equations
Aligned to A-REI.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
Solve rational equations by clearing denominators and radical equations by isolating the root and squaring. Because these steps can introduce false solutions, always check each answer in the original equation.
Worked examples
Example 1
Solve √(x + 2) = x. Squaring gives x + 2 = x^2, so x^2 − x − 2 = 0, which factors as (x − 2)(x + 1) = 0, giving x = 2 or x = −1. Checking: √4 = 2 works, but √1 = 1 ≠ −1, so x = −1 is extraneous and only x = 2 is valid.
Example 2
Solve 2/x + 3 = 5/x. Multiply every term by x: 2 + 3x = 5, so 3x = 3 and x = 1. Check that x = 1 does not make any denominator zero — it does not. Note also that x = 0 was never allowed, so it could never have been an answer.
Example 3
Common mistake: multiplying only some terms by the common denominator. In 2/x + 3 = 5/x, clearing the fractions means multiplying EVERY term, including the bare 3, which becomes 3x. Writing 2 + 3 = 5 instead gives the meaningless statement 5 = 5 and loses the solution entirely. Multiply across the whole equation, term by term, and nothing is left behind.
Practice questions
- Solve √x = 5.
- Solve √(2x − 1) = 3.
- Solve √(x + 6) = x and identify any extraneous solution.
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.
▶ Watch this lessonEn español
Ecuaciones racionales y con radicales
Resuelve ecuaciones racionales eliminando denominadores y ecuaciones con radicales aislando la raíz y elevando al cuadrado. Como estos pasos pueden introducir soluciones falsas, siempre verifica cada respuesta en la ecuación original.
Ejemplo 1: Resuelve √(x + 2) = x. Al elevar al cuadrado se obtiene x + 2 = x^2, así que x^2 − x − 2 = 0, que se factoriza como (x − 2)(x + 1) = 0, dando x = 2 o x = −1. Al verificar: √4 = 2 funciona, pero √1 = 1 ≠ −1, así que x = −1 es extraña y solo x = 2 es válida.
Ejemplo 2: Resuelve 2/x + 3 = 5/x. Multiplica cada término por x: 2 + 3x = 5, así que 3x = 3 y x = 1. Comprueba que x = 1 no anule ningún denominador — no lo hace. Observa además que x = 0 nunca estuvo permitido, así que jamás pudo ser respuesta.
Ejemplo 3: Error común: multiplicar solo algunos términos por el denominador común. En 2/x + 3 = 5/x, quitar las fracciones significa multiplicar TODOS los términos, incluido el 3 solitario, que se convierte en 3x. Escribir 2 + 3 = 5 da el enunciado vacío 5 = 5 y pierde la solución por completo. Multiplica toda la ecuación, término por término, y no se queda nada atrás.
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