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Giải các phương trình sau:
a. \(\left( {x – 1} \right)\left( {5x + 3} \right) = \left( {3x – 8} \right)\left( {x – 1} \right)\)
b. \(3x\left( {25x + 15} \right) – 35\left( {5x + 3} \right) = 0\)
c. \(\left( {2 – 3x} \right)\left( {x + 11} \right) = \left( {3x – 2} \right)\left( {2 – 5x} \right)\)
d. \(\left( {2{x^2} + 1} \right)\left( {4x – 3} \right) = \left( {2{x^2} + 1} \right)\left( {x – 12} \right)\)
e. \({\left( {2x – 1} \right)^2} + \left( {2 – x} \right)\left( {2x – 1} \right) = 0\)
f. \(\left( {x + 2} \right)\left( {3 – 4x} \right) = {x^2} + 4x + 4\)
a. \(\left( {x – 1} \right)\left( {5x + 3} \right) = \left( {3x – 8} \right)\left( {x – 1} \right)\)
\(\eqalign{ & \Leftrightarrow \left( {x – 1} \right)\left( {5x + 3} \right) – \left( {3x – 8} \right)\left( {x – 1} \right) = 0 \cr & \Leftrightarrow \left( {x – 1} \right)\left[ {\left( {5x + 3} \right) – \left( {3x – 8} \right)} \right] = 0 \cr & \Leftrightarrow \left( {x – 1} \right)\left( {5x + 3 – 3x + 8} \right) = 0 \cr & \Leftrightarrow \left( {x – 1} \right)\left( {2x + 11} \right) = 0 \cr} \)
\( \Leftrightarrow x – 1 = 0\)hoặc \(2x + 11 = 0\)
+ \(x – 1 = 0 \Leftrightarrow x = 1\)
+ \(2x + 11 = 0 \Leftrightarrow x = – 5,5\)
Phương trình có nghiệm x = 1 hoặc x = -5,5
b. \(3x\left( {25x + 15} \right) – 35\left( {5x + 3} \right) = 0\)
\(\eqalign{ & \Leftrightarrow 15x\left( {5x + 3} \right) – 35\left( {5x + 3} \right) = 0 \cr & \Leftrightarrow \left( {15x – 35} \right)\left( {5x + 3} \right) = 0 \cr} \)
\( \Leftrightarrow 15x – 35 = 0\) hoặc \(5x + 3 = 0\)
+ \(15x – 35 = 0 \Leftrightarrow x = {{35} \over {15}} = {7 \over 3}\)
+ \(5x + 3 = 0 \Leftrightarrow x = – {3 \over 5}\)
Phương trình có nghiệm \(x = {7 \over 3}\) hoặc \(x = – {3 \over 5}\)
c. \(\left( {2 – 3x} \right)\left( {x + 11} \right) = \left( {3x – 2} \right)\left( {2 – 5x} \right)\)
\(\eqalign{ & \Leftrightarrow \left( {2 – 3x} \right)\left( {x + 11} \right) – \left( {3x – 2} \right)\left( {2 – 5x} \right) = 0 \cr & \Leftrightarrow \left( {2 – 3x} \right)\left( {x + 11} \right) + \left( {2 – 3x} \right)\left( {2 – 5x} \right) = 0 \cr & \Leftrightarrow \left( {2 – 3x} \right)\left[ {\left( {x + 11} \right) + \left( {2 – 5x} \right)} \right] = 0 \cr & \Leftrightarrow \left( {2 – 3x} \right)\left( {x + 11 + 2 – 5x} \right) = 0 \cr & \Leftrightarrow \left( {2 – 3x} \right)\left( { – 4x + 13} \right) = 0 \cr} \)
\( \Leftrightarrow 2 – 3x = 0\)hoặc \(13 – 4x = 0\)
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+ \(2 – 3x = 0 \Leftrightarrow x = {2 \over 3}\)
+ \(13 – 4x = 0 \Leftrightarrow x = {{13} \over 4}\)
Phương trình có nghiệm \(x = {2 \over 3}\) hoặc \(x = {{13} \over 4}\)
d. \(\left( {2{x^2} + 1} \right)\left( {4x – 3} \right) = \left( {2{x^2} + 1} \right)\left( {x – 12} \right)\)
\(\eqalign{ & \Leftrightarrow \left( {2{x^2} + 1} \right)\left( {4x – 3} \right) – \left( {2{x^2} + 1} \right)\left( {x – 12} \right) = 0 \cr & \Leftrightarrow \left( {2{x^2} + 1} \right)\left[ {\left( {4x – 3} \right) – \left( {x – 12} \right)} \right] = 0 \cr & \Leftrightarrow \left( {2{x^2} + 1} \right)\left( {4x – 3 – x + 12} \right) = 0 \cr & \Leftrightarrow \left( {2{x^2} + 1} \right)\left( {3x + 9} \right) = 0 \cr} \)
\( \Leftrightarrow 2{x^2} + 1 = 0\)hoặc \(3x + 9 = 0\)
+ \(2{x^2} + 1 = 0\) vô nghiệm (\(2{x^2} \ge 0\) nên \(2{x^2} + 1 > 0$ )
+ \(3x + 9 = 0 \Leftrightarrow x = – 3\)
Phương trình có nghiệm x = -3
e. \({\left( {2x – 1} \right)^2} + \left( {2 – x} \right)\left( {2x – 1} \right) = 0\)
\(\eqalign{ & \Leftrightarrow \left( {2x – 1} \right)\left( {2x – 1} \right) + \left( {2 – x} \right)\left( {2x – 1} \right) = 0 \cr & \Leftrightarrow \left( {2x – 1} \right)\left[ {\left( {2x – 1} \right) + \left( {2 – x} \right)} \right] = 0 \cr & \Leftrightarrow \left( {2x – 1} \right)\left( {2x – 1 + 2 – x} \right) = 0 \cr & \Leftrightarrow \left( {2x – 1} \right)\left( {x + 1} \right) = 0 \cr} \)
\( \Leftrightarrow 2x – 1 = 0\)hoặc \(x + 1 = 0\)
+ \(2x – 1 = 0 \Leftrightarrow x = 0,5\)
+ \(x + 1 = 0 \Leftrightarrow x = – 1\)
Phương trình có nghiệm x = 0,5 hoặc x = -1
f. \(\left( {x + 2} \right)\left( {3 – 4x} \right) = {x^2} + 4x + 4\)
\(\eqalign{ & \Leftrightarrow \left( {x + 2} \right)\left( {3 – 4x} \right) – {\left( {x + 2} \right)^2} = 0 \cr & \Leftrightarrow \left( {x + 2} \right)\left( {3 – 4x} \right) – \left( {x + 2} \right)\left( {x + 2} \right) = 0 \cr & \Leftrightarrow \left( {x + 2} \right)\left[ {\left( {3 – 4x} \right) – \left( {x + 2} \right)} \right] = 0 \cr & \Leftrightarrow \left( {x + 2} \right)\left( {3 – 4x – x – 2} \right) = 0 \cr & \Leftrightarrow \left( {x + 2} \right)\left( {1 – 5x} \right) = 0 \cr} \)
\( \Leftrightarrow x + 2 = 0\) hoặc \(1 – 5x = 0\)
+ \(x + 2 = 0 \Leftrightarrow x = – 2\)
+ \(1 – 5x = 0 \Leftrightarrow x = 0,2\)
Phương trình có nghiệm x = -2 hoặc x = 0,2