Name: solving radical equations with checks
Uploaded: 2025-11-26T05:44:27.000Z
Duration: 1 min 26 s
Channel: JoVE

Le equazioni dei radicali sono equazioni matematiche in cui la variabile appare all'interno di un radicale, come una radice quadrata, una radice cubica o una quarta radice.Risolverli comporta l'eliminazione del radicale elevando entrambi i lati dell'equazione a una potenza pari all'indice del radicale.Ad esempio, calcolare la profondità di un pozzo facendo cadere un oggetto e misurando il tempo fino a quando non si sente lo schizzo è un'applicazione pratica di un'equazione radicale.Qui, il tempo totale include il tempo di caduta, dato dalla radice quadrata del doppio della profondità rispetto all'accelerazione gravitazionale, e il tempo di viaggio del suono, dato dalla profondità rispetto alla velocità del suono, formando un'equazione radicale.Il primo passo è isolare il termine radicale e poi elevare al quadrato entrambi i lati per eliminare la radice quadrata.Questo passaggio trasforma l'equazione radicalica in un'equazione quadratica standard, che può essere ulteriormente semplificata. La formula quadratica viene applicata per risolverlo, fornendo due possibili valori per la profondità.Ogni soluzione viene sostituita nell'equazione radicale originale per verificare se soddisfa veramente l'equazione o se è una radice estranea.Solo una soluzione fornisce un numero valido e positivo per la profondità, confermandola come la soluzione corretta.

equazioni radicali

Le equazioni radicali sono equazioni matematiche in cui la variabile appare all'interno di un radicale, pi&#249; comunemente la radice quadrata o cubica. Queste equazioni si presentano frequentemente in ambito scientifico e ingegneristico, nonch&#233; nelle misurazioni del mondo reale che coinvolgono relazioni non lineari. Per risolvere un'equazione radicale, la procedura standard consiste nell'isolare l'espressione radicale e quindi eliminare il radicale elevando entrambi i membri a una potenza pari all'indice del radicale. Questo processo pu&#242; portare a soluzioni spurie, cio&#232; valori che risolvono l'equazione trasformata ma non quella originale. Pertanto, tutte le soluzioni devono essere verificate per sostituzione.Determinare le soluzioni reali dell'equazione: &#8730;(3x+1) + 2 = x<ul><li>Isolare il radicale:</li></ul><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:524/80;' src='https://cloudfront.jove.com/files/ftp_upload/40034/40034_Equation_1_editor_40034.svg' alt='40034_Equation_1.svg' /></figure><ul><li>Elevare entrambi i membri al quadrato (eliminando la radice quadrata):</li></ul><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:802/183;' src='https://cloudfront.jove.com/files/ftp_upload/40034/40034_Equation_2_editor_40034.svg' alt='40034_Equation_2.svg' /></figure><ul><li>Portare tutti i termini su un solo membro per ottenere un'equazione di secondo grado standard:</li></ul><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:494/74;' src='https://cloudfront.jove.com/files/ftp_upload/40034/40034_Equation_3_editor_40034.svg' alt='40034_Equation_3.svg' /></figure><ul><li>Risolvere l'equazione di secondo grado usando la formula risolutiva:</li></ul><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1905/449;' src='https://cloudfront.jove.com/files/ftp_upload/40034/40034_Equation_4_editor_40034.svg' alt='40034_Equation_4.svg' /></figure><ul><li>Verificare la presenza di soluzioni spurie: solo x &#8776; 6,54 soddisfa l'equazione originale. Sostituendo x &#8776; 0,46 nell'espressione &#8730;(3x+1), si ottiene un valore positivo, mentre l'espressione x - 2 diventa negativa. Poich&#233; la radice quadrata &#232; definita per produrre un risultato non negativo e non pu&#242; essere uguale a un numero negativo, x &#8776; 0,46 &#232; una soluzione spuria.</li></ul>

Equazioni radicali

Le equazioni radicali sono equazioni matematiche in cui la variabile appare all'interno di un radicale, pi&#249; comunemente la radice quadrata o cubica. ...

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Radical Equations

solving radical equations with checks

Radical equations are equations with a variable inside a radical, most often a square root or cube root. These equations appear in science, engineering, and real-world measurement problems that involve nonlinear relationships. Solving them starts by isolating the radical expression. Then the radical is removed by raising both sides to the power that matches the radical’s index.

This process can create extraneous solutions, which are values that work in the transformed equation but not in the original one. Because of that, every possible solution must be checked by substitution. The example in the transcript begins with 3 x + 1 + 2 = x 1. After isolating the radical, both sides are squared to eliminate the square root.

The equation is then rewritten as a standard quadratic equation. From there, the quadratic formula is used to find the possible solutions. Only x ≈ 6.54 satisfies the original equation. The other value, x ≈ 0.46, is extraneous because 3 x + 1 stays positive while x - 2 becomes negative. A square root must give a nonnegative result, so it cannot equal a negative number.

Solving Radical Equations with Checks

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radical equations

Radical equations are equations with a variable inside a radical, most often a square root or cube root. These equations appear in science, engineering, and real-world measurement problems that involve nonlinear relationships. Solving them starts by isolating the radical expression. Then the radical is removed by raising both sides to the power that matches the radical’s index. This process can create extraneous solutions, which are values that work in the transformed equation but not in the...

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key terms and concepts

Learn how to isolate a radical, square both sides, and solve the resulting quadratic equation while checking for extraneous solutions.

Solving Radical Equations with Checks | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

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Solving Radical Equations with Checks

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