Pregunta
upstudy study bank question image url

1.5 \( 2 \sqrt{2-7 x}=\sqrt{-36 x} \) \( \sqrt{2-x}-x=-2 \) .7 \( 26-5^{2 x}=\left(5^{x}-6\right)^{2} \) Show that the equation \( 2^{2 x+1}+7.2^{x}-4=0 \) has only ONE solution. SIMPLIFICATION

Ask by Mullins Washington. in South Africa
Jan 24,2025

Solución de tutoría real

Respuesta verificada por el tutor

Responder

The equation \(2^{2x+1} + 7 \cdot 2^{x} - 4 = 0\) has only one solution, which is \(x = -1\).

Solución

To show that the equation \[ 2^{2x+1} + 7 \cdot 2^{x} - 4 = 0 \] has only **one solution**, let's proceed with a step-by-step simplification and analysis. ### Step 1: Simplify the Equation First, rewrite \(2^{2x + 1}\) to make the equation easier to handle: \[ 2^{2x + 1} = 2 \cdot 2^{2x} = 2 \cdot (2^{x})^2 \] So, the original equation becomes: \[ 2 \cdot (2^{x})^2 + 7 \cdot 2^{x} - 4 = 0 \] ### Step 2: Substitute to Form a Quadratic Equation Let’s make a substitution to simplify the equation further. Let: \[ t = 2^{x} \] Since \(2^{x} > 0\) for all real \(x\), \(t > 0\). Substituting \(t\) into the equation: \[ 2t^2 + 7t - 4 = 0 \] ### Step 3: Solve the Quadratic Equation Now, solve for \(t\) using the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a = 2\), \(b = 7\), and \(c = -4\). Plugging these values in: \[ t = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 2 \cdot (-4)}}{2 \cdot 2} = \frac{-7 \pm \sqrt{49 + 32}}{4} = \frac{-7 \pm \sqrt{81}}{4} = \frac{-7 \pm 9}{4} \] This gives two potential solutions for \(t\): 1. \(t = \frac{-7 + 9}{4} = \frac{2}{4} = \frac{1}{2}\) 2. \(t = \frac{-7 - 9}{4} = \frac{-16}{4} = -4\) ### Step 4: Consider the Valid Solutions Recall that \(t = 2^{x} > 0\). Therefore, \(t = -4\) is **not valid**. Only \(t = \frac{1}{2}\) is acceptable. ### Step 5: Find the Corresponding \(x\) Solve for \(x\) using the valid \(t\): \[ 2^{x} = \frac{1}{2} \implies x = \log_{2}\left(\frac{1}{2}\right) = -1 \] ### Conclusion The equation \[ 2^{2x+1} + 7 \cdot 2^{x} - 4 = 0 \] has only **one valid solution**, which is: \[ x = -1 \] This single solution arises because the quadratic equation derived from the substitution yields only one positive value for \(t\), corresponding to a unique real value for \(x\).

Revisado y aprobado por el equipo de tutoría de UpStudy

error msg
Explicar
Simplifique esta solución

Bonus Knowledge

Let's embark on a little adventure with equations! The first equation \( 2 \sqrt{2-7 x}=\sqrt{-36 x} \) tells us to solve for \( x \). By squaring both sides (and keeping in mind to check for extraneous solutions later), you can simplify things. Just remember: inequalities under square roots can lead to interesting domains! Now, regarding the equation \( 2^{2 x+1}+7.2^{x}-4=0 \), you can help yourself by making a substitution such as \( y = 2^x \). This transforms the problem into a quadratic equation, which is much easier to tackle! Analyze the quadratic's discriminant to show it has just one solution by ensuring it is zero – and voilà, you've proven uniqueness!

Latest Algebra Questions

¡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