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

Wurzelgleichungen sind mathematische Gleichungen, in denen die Variable innerhalb einer Wurzel vorkommt, z. B. einer Quadratwurzel, einer Kubikwurzel oder einer vierten Wurzel.Um sie zu lösen, muss das Radikal eliminiert werden, indem beide Seiten der Gleichung auf eine Potenz angehoben werden, die dem Index des Radikals entspricht.Zum Beispiel ist die Berechnung der Tiefe eines Bohrlochs, indem man ein Objekt fallen lässt und die Zeit misst, bis das Plätschern zu hören ist, eine praktische Anwendung einer radikalen Gleichung.Hier umfasst die Gesamtzeit die Fallzeit, gegeben durch die Quadratwurzel der doppelten Tiefe über der Gravitationsbeschleunigung, und die Schalllaufzeit, gegeben durch die Tiefe über die Schallgeschwindigkeit, wodurch eine Radikalgleichung gebildet wird.Der erste Schritt besteht darin, den Radikalterm zu isolieren und dann beide Seiten zu quadrieren, um die Quadratwurzel zu eliminieren.In diesem Schritt wird die Radikalgleichung in eine quadratische Standardgleichung umgewandelt, die weiter vereinfacht werden kann. Zur Lösung wird die quadratische Formel angewendet, die zwei mögliche Werte für die Tiefe ergibt.Jede Lösung wird wieder in die ursprüngliche Wurzelgleichung eingesetzt, um zu prüfen, ob sie die Gleichung wirklich erfüllt oder ob es sich um eine fremde Wurzel handelt.Nur eine Lösung liefert eine gültige, positive Zahl für die Tiefe, die bestätigt, dass sie die richtige Lösung ist.

wurzelgleichungen

Wurzelgleichungen sind Gleichungen, in denen die Variable unter einem Wurzelzeichen steht, meist unter einer Quadratwurzel oder Kubikwurzel. Solche Gleichungen treten h&#228;ufig in Wissenschaft und Technik sowie bei realen Messungen mit nichtlinearen Zusammenh&#228;ngen auf. Zur L&#246;sung wird der Wurzelausdruck zun&#228;chst isoliert und anschlie&#223;end durch Potenzieren beider Seiten mit dem Index der Wurzel eliminiert. Dieses Vorgehen kann zu Scheinl&#246;sungen f&#252;hren &#8211; Werten, die die transformierte Gleichung erf&#252;llen, jedoch nicht die urspr&#252;ngliche &#8211;, daher m&#252;ssen alle m&#246;glichen L&#246;sungen durch Einsetzen verifiziert werden.Bestimmen Sie die reelle(n) L&#246;sung(en) der Gleichung: &#8730;(3x+1)+2=x<ul><li>Isolieren Sie den Wurzelausdruck:</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>Quadrieren Sie beide Seiten (Eliminierung der Quadratwurzel):</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>Bringen Sie alle Terme auf eine Seite, um eine quadratische Gleichung in Normalform zu erhalten:</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>L&#246;sen Sie die quadratische Gleichung mit der quadratischen Formel:</li></ul><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:65.05%;'><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>Pr&#252;fen Sie auf Scheinl&#246;sungen: Nur x &#8776; 6,54 erf&#252;llt die urspr&#252;ngliche Gleichung. Setzt man x &#8776; 0,46 ein, ergibt &#8730;(3x+1) einen positiven Wert, w&#228;hrend x&#8722;2 negativ ist. Da die Quadratwurzel definitionsgem&#228;&#223; nichtnegativ ist und daher nicht gleich einer negativen Zahl sein kann, ist x &#8776; 0,46 eine Scheinl&#246;sung.</li></ul>

Wurzelgleichungen

Wurzelgleichungen sind Gleichungen, in denen die Variable unter einem Wurzelzeichen steht, meist unter einer Quadratwurzel oder Kubikwurzel. Solche ...

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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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