Advanced Equation Solving

This page continues with Question 5, which involves solving various types of quadratic and related equations.

a) 6x² = 6144 is solved to yield x = ±32. b) x² - 6x - 7 = 0 is solved using the quadratic formula, resulting in x = 7 or x = -1. c) 3x² + 17 = -(x + 1)(x - 1) is transformed and solved, showing that it has no real solutions. d) (2x-9)(5x+3) = 0 is solved using the zero product property, giving x = 4.5 or x = -0.6.

Highlight: The solution notes that using the zero product property for part d) would have been simpler than expanding the equation.

e) -3x² - 4x = 0 is factored and solved, resulting in x = 0 or x = -4/3.

These problems demonstrate various techniques for solving quadratische Gleichungen (quadratic equations), including factoring, using the quadratic formula, and applying algebraic properties.

Word Problems and Practical Applications

The final page of the exam focuses on Question 6, which presents two word problems applying quadratic equations to real-world scenarios.

a) The first problem involves finding a number that, when decreased by 5 and multiplied by itself plus 2, equals 408. This is solved by setting up the equation (x - 5)(x + 2) = 408 and solving it to find x = 22.

b) The second problem deals with calculating the dimensions of a cuboid given its volume and height, and a relationship between its length and width. The problem is solved step-by-step:

• Volume: 765 cm³
• Height: 17 cm
• Length = Width + 4 cm

Example: The solution involves setting up the equation (y + 4) · y · 17 = 765, where y represents the width of the cuboid. Solving this quadratic equation yields y = 5 cm (width) and x = 9 cm (length).

This question demonstrates the practical application of quadratische Gleichungen Aufgaben (quadratic equation problems) in real-world contexts, reinforcing the importance of these mathematical concepts.

Mathematics Exam Overview

This page presents a mathematics exam for Class 8a, covering various topics related to roots and quadratic equations. The exam is structured with six main questions, totaling 26 points.

Highlight: The exam emphasizes the importance of clear presentation, stating that poor presentation may result in point deductions.

The questions cover the following areas:

1. Determining definition sets for root terms
2. Simplifying root expressions and rationalizing denominators
3. Identifying quadratic equations from graphs
4. Solving quadratic equations using the completing the square method
5. Solving various forms of quadratic and related equations
6. Applying quadratic equations to word problems

Vocabulary: Definitionsmenge (Definition set) refers to the set of all possible input values for a mathematical function, which is particularly important when dealing with root terms.

Soluciones y Evaluación

Las páginas finales del documento muestran las soluciones detalladas para cada problema de la prueba. Estas soluciones proporcionan un valioso recurso para que los estudiantes comparen sus respuestas y comprendan los métodos correctos de resolución.

Vocabulario: Lösungsmenge (conjunto solución) - El conjunto de todos los valores que satisfacen una ecuación o sistema de ecuaciones.

La evaluación final muestra una puntuación de 24/26, indicando un excelente desempeño en la prueba. Esto sugiere que el estudiante tiene un buen dominio de los conceptos de raíces cuadradas y ecuaciones cuadráticas, áreas que son fundamentales en matemáticas de nivel secundario y preparatorio para estudios más avanzados.

Highlight: Para una preparación completa, los estudiantes deberían practicar con Quadratische Gleichungen Aufgaben mit Lösungen Klasse 9 PDF y Wurzeln Aufgaben Klasse 9 PDF.