giải pt

Question

1. $ \left ( \sqrt{4-\sqrt{15}}\right )^{x} + \left ( \sqrt{4+\sqrt{15}} \right )^{x} = (2\sqrt{2})^{x}$
2. $ (5-\sqrt{21})^{x} +7 (5+\sqrt{21})^x= 2^{x+3} $

bạn xem lại đề câu 2 xem có mũ x ở bt sau k nhá.

score 8 · Answer 1

Bình chọn tăng 8 Bình chọn giảm

1)
Ta có:
$\left(\sqrt{4-\sqrt{15}}\right)^x+\left(\sqrt{4+\sqrt{15}}\right)^x=(2\sqrt2)^x$
$\Leftrightarrow \left(\sqrt{8-2\sqrt{15}}\right)^x+\left(\sqrt{8+2\sqrt{15}}\right)^x=4^x$
$\Leftrightarrow (\sqrt5-\sqrt3)^x+(\sqrt5+\sqrt3)^x=4^x$
$\Leftrightarrow \left(\frac{\sqrt5-\sqrt3}{4}\right)^x+\left(\frac{\sqrt5+\sqrt3}{4}\right)^x=1$
Xét hàm: $f(x)=\left(\frac{\sqrt5-\sqrt3}{4}\right)^x+\left(\frac{\sqrt5+\sqrt3}{4}\right)^x$
Ta có:
$f'(x)=\ln\left(\frac{\sqrt5-\sqrt3}{4}\right)\left(\frac{\sqrt5-\sqrt3}{4}\right)^x+\ln\left(\frac{\sqrt5+\sqrt3}{4}\right)\left(\frac{\sqrt5+\sqrt3}{4}\right)^x<0,\forall x\in\mathbb{R}$
$\Rightarrow f(x)$ nghịch biến trên $\mathbb{R}$.
Suy ra: $f(x)=1$ có nhiều nhất 1 nghiệm.
Mà $f(2)=1$.
Vậy phương trình có nghiệm duy nhất: $x=2$.

like phát – gaara.sshn 19-11-12 08:56 PM

chuẩn rồi đấy – kellyhoang297 19-11-12 08:39 PM

thanks nha – luffykunneu 19-11-12 08:25 PM

thank, Bài 2 – conan95_iloveyou 19-11-12 02:10 PM

score 5 · Answer 2

Bình chọn tăng 5 Bình chọn giảm

2.
Phương trình tương đương với:
$\left(\frac{5-\sqrt{21}}{2}\right)^x+7\left(\frac{5+\sqrt{21}}{2}\right)^x=8$
Đặt: $a=\left(\frac{5-\sqrt{21}}{2}\right)^x,b=\left(\frac{5+\sqrt{21}}{2}\right)^x;a,b>0$
Ta có hệ sau:
$\left\{ \begin{array}{l} ab=1\\a+7b=8 \end{array} \right.$
$\Leftrightarrow \left\{ \begin{array}{l} b=\frac{1}{a}\\a+\frac{7}{a}=8 \end{array} \right.$
$\Leftrightarrow \left[ \begin{array}{l} \left\{ \begin{array}{l} a=1\\b=1 \end{array} \right.\\\left\{ \begin{array}{l} a=7\\ b=\frac{1}{7} \end{array} \right. \end{array} \right.$
Suy ra:
$\left[ \begin{array}{l}\left(\frac{5-\sqrt{21}}{2}\right)^x=1 \\\left(\frac{5-\sqrt{21}}{2}\right)^x=7\end{array} \right.\Leftrightarrow \left[\begin{array}{l} x=0\\x=\frac{\ln7}{\ln(5-\sqrt{21})-\ln2} \end{array} \right.$

Nếu thấy lời giải đúng thì bạn vui lòng đánh dấu vào hình chữ V dưới phần vote để xác nhận nhá. Thanks – khangnguyenthanh 05-01-13 09:04 PM