Pregunta
upstudy study bank question image url

What is the difference? \( \frac{2 x+5}{x^{2}-3 x}-\frac{3 x+5}{x^{3}-9 x}-\frac{x+1}{x^{2}-9} \)

Ask by Gibson Lyons. in the United States
Mar 11,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The difference is \( \frac{x^{2}+7x+10}{x^{3}-9x} \).

Solución

Calculate or simplify the expression \( \frac{2x+5}{x^2-3x}-\frac{3x+5}{x^3-9x}-\frac{x+1}{x^2-9} \). Simplify the expression by following steps: - step0: Solution: \(\frac{2x+5}{x^{2}-3x}-\frac{3x+5}{x^{3}-9x}-\frac{x+1}{x^{2}-9}\) - step1: Factor the expression: \(\frac{2x+5}{x^{2}-3x}-\frac{3x+5}{\left(x+3\right)\left(x^{2}-3x\right)}-\frac{x+1}{\left(x-3\right)\left(x+3\right)}\) - step2: Reduce fractions to a common denominator: \(\frac{\left(2x+5\right)\left(x+3\right)}{\left(x^{2}-3x\right)\left(x+3\right)}-\frac{3x+5}{\left(x+3\right)\left(x^{2}-3x\right)}-\frac{\left(x+1\right)x}{\left(x-3\right)\left(x+3\right)x}\) - step3: Rewrite the expression: \(\frac{\left(2x+5\right)\left(x+3\right)}{\left(x+3\right)\left(x^{2}-3x\right)}-\frac{3x+5}{\left(x+3\right)\left(x^{2}-3x\right)}-\frac{\left(x+1\right)x}{\left(x+3\right)\left(x^{2}-3x\right)}\) - step4: Transform the expression: \(\frac{\left(2x+5\right)\left(x+3\right)-\left(3x+5\right)-\left(x+1\right)x}{\left(x+3\right)\left(x^{2}-3x\right)}\) - step5: Multiply the terms: \(\frac{2x^{2}+11x+15-\left(3x+5\right)-\left(x+1\right)x}{\left(x+3\right)\left(x^{2}-3x\right)}\) - step6: Multiply the terms: \(\frac{2x^{2}+11x+15-\left(3x+5\right)-\left(x^{2}+x\right)}{\left(x+3\right)\left(x^{2}-3x\right)}\) - step7: Calculate: \(\frac{x^{2}+7x+10}{\left(x+3\right)\left(x^{2}-3x\right)}\) - step8: Multiply the terms: \(\frac{x^{2}+7x+10}{x^{3}-9x}\) The difference between the given expressions is \( \frac{x^{2}+7x+10}{x^{3}-9x} \).

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Mind Expander

To simplify the expression, let's first identify and factor the denominators. 1. The first denominator \( x^2 - 3x \) can be factored as \( x(x - 3) \). 2. The second denominator \( x^3 - 9x \) can be factored by taking out \( x \), giving us \( x(x^2 - 9) \). The expression \( x^2 - 9 \) is a difference of squares, which factors to \( x(x - 3)(x + 3) \). 3. The third denominator \( x^2 - 9 \) also factors as \( (x - 3)(x + 3) \). Now, let's write the common denominator. The least common denominator (LCD) will be \( x(x - 3)(x + 3) \). Now we rewrite each fraction with the common denominator: 1. For the first fraction: \[ \frac{2x + 5}{x^2 - 3x} = \frac{2x + 5}{x(x - 3)} \times \frac{(x + 3)}{(x + 3)} = \frac{(2x + 5)(x + 3)}{x(x - 3)(x + 3)} \] 2. For the second fraction: \[ \frac{3x + 5}{x^3 - 9x} = \frac{3x + 5}{x(x - 3)(x + 3)} \quad \text{(it is already in the appropriate form)} \] 3. For the third fraction: \[ \frac{x + 1}{x^2 - 9} = \frac{x + 1}{(x - 3)(x + 3)} \times \frac{x}{x} = \frac{x(x + 1)}{x(x - 3)(x + 3)} \] Now combine all three fractions: \[ \frac{(2x + 5)(x + 3)-(3x + 5) - x(x + 1)}{x(x - 3)(x + 3)} \] Next, simplify the numerator: 1. Expand \( (2x + 5)(x + 3) \): \[ = 2x^2 + 6x + 5x + 15 = 2x^2 + 11x + 15 \] Now combine everything in the numerator: \[ 2x^2 + 11x + 15 - (3x + 5) - (x^2 + x) \] Distributing the minus signs gives: \[ 2x^2 + 11x + 15 - 3x - 5 - x^2 - x \] Combining like terms: \[ (2x^2 - x^2) + (11x - 3x - x) + (15 - 5) = x^2 + 7x + 10 \] Now our expression is: \[ \frac{x^2 + 7x + 10}{x(x - 3)(x + 3)} \] Finally, we can factor the numerator: \[ x^2 + 7x + 10 = (x + 2)(x + 5) \] Thus, the final simplified form is: \[ \frac{(x + 2)(x + 5)}{x(x - 3)(x + 3)} \] Just remember, this expression is valid as long as \( x \neq 0 \), \( x \neq 3 \), and \( x \neq -3 \) to avoid division by zero.

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad