Bài 5: Công thức nghiệm thu gọn
Hướng dẫn giải Bài 18 (Trang 49 SGK Toán Đại số 9, Tập 2)
<p>Đưa c&aacute;c phương tr&igrave;nh sau về dạng ax<sup>2</sup> + 2b'x + c = 0 v&agrave; giải ch&uacute;ng. Sau đ&oacute;, d&ugrave;ng bảng số hoặc m&aacute;y t&iacute;nh để viết gần đ&uacute;ng nghiệm t&igrave;m được (l&agrave;m tr&ograve;n kết quả đến chữ số thập ph&acirc;n thứ hai):</p> <p>&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi><mo>)</mo><mo>&#160;</mo><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>=</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mi mathvariant="normal">b</mi><mo>)</mo><mo>&#160;</mo><msup><mrow><mo>(</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><msqrt><mn>2</mn></msqrt><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>;</mo><mspace linebreak="newline"/><mi mathvariant="normal">c</mi><mo>)</mo><mo>&#160;</mo><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>=</mo><mn>2</mn><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mi mathvariant="normal">d</mi><mo>)</mo><mo>&#160;</mo><mn>0</mn><mo>,</mo><mn>5</mn><mi mathvariant="normal">x</mi><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup><mo>.</mo></math></p> <p><strong>Giải:</strong></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi><mo>)</mo><mo>&#160;</mo><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>=</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>&#8660;</mo><mn>2</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>0</mn><mo>.</mo><mspace linebreak="newline"/><mi mathvariant="normal">b</mi><mo>'</mo><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo>&#160;</mo><mo>&#8710;</mo><mo>'</mo><mo>=</mo><msup><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>.</mo><mo>(</mo><mo>-</mo><mn>3</mn><mo>)</mo><mo>=</mo><mn>7</mn><mspace linebreak="newline"/><msub><mi mathvariant="normal">x</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><msqrt><mn>7</mn></msqrt></mrow><mn>2</mn></mfrac><mo>&#8776;</mo><mn>1</mn><mo>,</mo><mn>82</mn><mo>;</mo><mo>&#160;</mo><msub><mi mathvariant="normal">x</mi><mn>2</mn></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><msqrt><mn>7</mn></msqrt></mrow><mn>2</mn></mfrac><mo>&#8776;</mo><mo>-</mo><mn>0</mn><mo>,</mo><mn>82</mn><mspace linebreak="newline"/><mspace linebreak="newline"/><mi mathvariant="normal">b</mi><mo>)</mo><mo>&#160;</mo><msup><mrow><mo>(</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><msqrt><mn>2</mn></msqrt><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>&#8660;</mo><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msqrt><mn>2</mn></msqrt><mo>.</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn><mo>.</mo><mspace linebreak="newline"/><mi mathvariant="normal">b</mi><mo>'</mo><mo>=</mo><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>,</mo><mo>&#160;</mo><mo>&#8710;</mo><mo>'</mo><mo>=</mo><msup><mrow><mo>(</mo><mo>-</mo><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mn>3</mn><mo>.</mo><mn>2</mn><mo>=</mo><mn>2</mn><mspace linebreak="newline"/><msub><mi mathvariant="normal">x</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>+</mo><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>&#8776;</mo><mn>1</mn><mo>,</mo><mn>41</mn><mo>;</mo><mo>&#160;</mo><msub><mi mathvariant="normal">x</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>-</mo><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac><mo>=</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>3</mn></mfrac><mo>&#8776;</mo><mn>0</mn><mo>,</mo><mn>47</mn><mspace linebreak="newline"/><mspace linebreak="newline"/><mi mathvariant="normal">c</mi><mo>)</mo><mo>&#160;</mo><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>=</mo><mn>2</mn><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>&#8660;</mo><mn>3</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn><mo>.</mo><mspace linebreak="newline"/><mi mathvariant="normal">b</mi><mo>'</mo><mo>=</mo><mn>1</mn><mo>;</mo><mo>&#160;</mo><mo>&#8710;</mo><mo>'</mo><mo>=</mo><msup><mrow><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mn>3</mn><mo>.</mo><mn>1</mn><mo>=</mo><mo>-</mo><mn>2</mn><mo>&#60;</mo><mn>0</mn><mspace linebreak="newline"/><mi>Ph&#432;&#417;ng</mi><mo>&#160;</mo><mi>tr&#236;nh</mi><mo>&#160;</mo><mi>v&#244;</mi><mo>&#160;</mo><mi>ngh&#7879;m</mi><mo>.</mo><mspace linebreak="newline"/><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>)</mo><mo>&#160;</mo><mn>0</mn><mo>,</mo><mn>5</mn><mi mathvariant="normal">x</mi><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup><mo>&#8660;</mo><mn>0</mn><mo>,</mo><mn>5</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>,</mo><mn>5</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn><mspace linebreak="newline"/><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#8660;</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn><mo>,</mo><mo>&#160;</mo><mi mathvariant="normal">b</mi><mo>'</mo><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>;</mo><mo>&#160;</mo><mo>&#8710;</mo><mo>'</mo><mo>=</mo><msup><mrow><mo>(</mo><mo>-</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>.</mo><mn>2</mn><mo>=</mo><mn>4</mn><mo>,</mo><mn>25</mn><mspace linebreak="newline"/><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><mo>&#160;</mo><msub><mi mathvariant="normal">x</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>+</mo><msqrt><mn>4</mn><mo>,</mo><mn>25</mn></msqrt><mo>&#8776;</mo><mn>4</mn><mo>,</mo><mn>56</mn><mo>;</mo><mo>&#160;</mo><msub><mi mathvariant="normal">x</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>+</mo><msqrt><mn>4</mn><mo>,</mo><mn>25</mn></msqrt><mo>&#8776;</mo><mn>0</mn><mo>,</mo><mn>44</mn></math></p> <p>(R&otilde; r&agrave;ng trong trường hợp n&agrave;y d&ugrave;ng c&ocirc;ng thức nghiệm thu gọn cũng kh&ocirc;ng đơn giản hơn)</p>
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