Giải bài 41 trang 60 SBT toán 10 - Cánh diều

Đề bài

Giải các phương trình sau:

a) \(\sqrt {7 - 2x}  + x = 2\)

b) \(\sqrt { - 2{x^2} + 7x + 1}  + 3x = 7\)

Lời giải chi tiết

a) \(\sqrt {7 - 2x}  + x = 2 \Leftrightarrow \sqrt {7 - 2x}  = 2 - x\)

\(\begin{array}{l} \Leftrightarrow \left\{ \begin{array}{l}2 - x \ge 0\\7 - 2x = {\left( {2 - x} \right)^2}\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x \le 2\\7 - 2x = {x^2} - 4x + 4\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}x \le 2\\{x^2} - 2x - 3 = 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x \le 2\\\left[ \begin{array}{l}x = 3\;(L)\\x =  - 1\;\end{array} \right.\end{array} \right.\end{array}\)

Vậy \(S = \left\{ { - 1} \right\}\)

b) \(\sqrt { - 2{x^2} + 7x + 1}  + 3x = 7 \Leftrightarrow \sqrt { - 2{x^2} + 7x + 1}  = 7 - 3x\)

\(\begin{array}{l} \Leftrightarrow \left\{ \begin{array}{l}7 - 3x \ge 0\\ - 2{x^2} + 7x + 1 = {\left( {7 - 3x} \right)^2}\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x \le \frac{7}{3}\\ - 2{x^2} + 7x + 1 = 9{x^2} - 42x + 49\end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l}x \le \frac{7}{3}\\11{x^2} - 49x + 48 = 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x \le \frac{7}{3}\\\left[ \begin{array}{l}x = 3\;(L)\\x = \frac{{16}}{{11}}\;\end{array} \right.\quad \end{array} \right. \Leftrightarrow x = \frac{{16}}{{11}}\;\end{array}\)

Vậy \(S = \left\{ {\frac{{16}}{{11}}} \right\}\)