Giải các phương trình:
a) \({\left( {x + 2} \right)^2} – 3x – 5 = \left( {1 – x} \right)\left( {1 + x} \right)\)
b) \({\left( {x – 1} \right)^3} + 2x = {x^3} – {x^2} – 2x + 1\)
c) \(x\left( {{x^2} – 6} \right) – {\left( {x – 2} \right)^2} = {\left( {x + 1} \right)^3}\)
d) \({\left( {x + 5} \right)^2} + {\left( {x – 2} \right)^2} + \left( {x + 7} \right)\left( {x – 7} \right) = 12x – 23\)
a)
\(\eqalign{
& {\left( {x + 2} \right)^2} – 3x – 5 = \left( {1 – x} \right)\left( {1 + x} \right) \cr
& \Leftrightarrow {x^2} + 4x + 4 – 3x – 5 = 1 – {x^2} \cr
& \Leftrightarrow 2{x^2} + x – 2 = 0 \cr
& \Delta = 1 – 4.2.\left( { – 2} \right) = 1 + 16 = 17 > 0 \cr
& \sqrt \Delta = \sqrt {17} \cr
& {x_1} = {{ – 1 + \sqrt {17} } \over {2.2}} = {{\sqrt {17} – 1} \over 4} \cr
& {x_2} = {{ – 1 – \sqrt {17} } \over {2.2}} = – {{1 + \sqrt {17} } \over 4} \cr} \)
Advertisements (Quảng cáo)
b)
\(\eqalign{
& {\left( {x – 1} \right)^3} + 2x = {x^3} – {x^2} – 2x + 1 \cr
& \Leftrightarrow {x^3} – 3{x^2} + 3x – 1 + 2x = {x^3} – {x^2} – 2x + 1 \cr
& \Leftrightarrow 2{x^2} – 7x + 2 = 0 \cr
& \Delta = {\left( { – 7} \right)^2} – 4.2.2 = 49 – 16 = 33 > 0 \cr
& \sqrt \Delta = \sqrt {33} \cr
& {x_1} = {{7 + \sqrt {33} } \over {2.2}} = {{7 + \sqrt {33} } \over 4} \cr
& {x_2} = {{7 – \sqrt {33} } \over {2.2}} = {{7 – \sqrt {33} } \over 4} \cr} \)
c)
\(\eqalign{
& x\left( {{x^2} – 6} \right) – {\left( {x – 2} \right)^2} = {\left( {x + 1} \right)^3} \cr
& \Leftrightarrow {x^3} – 6x – {x^2} + 4x – 4 = {x^3} + 3{x^2} + 3x + 1 \cr
& \Leftrightarrow 4{x^2} + 5x + 5 = 0 \cr
& \Delta = {5^2} – 4.4.5 = 25 – 80 = – 55 < 0 \cr} \)
Phương trình vô nghiệm.
d)
\(\eqalign{
& {\left( {x + 5} \right)^2} + {\left( {x – 2} \right)^2} + \left( {x + 7} \right)\left( {x – 7} \right) = 12x – 23 \cr
& \Leftrightarrow {x^2} + 10x + 25 + {x^2} – 4x + 4 + {x^2} – 49 – 12x + 23 = 0 \cr
& \Leftrightarrow 3{x^2} – 6x + 3 = 0 \cr
& \Leftrightarrow {x^2} – 2x + 1 = 0 \cr
& \Delta ‘ = {1^2} – 1.1 = 1 – 1 = 0 \cr} \)
Phương trình có nghiệm số kép: \({x_1} = {x_2} = 1\)