Bài 5: Công thức nghiệm thu gọn
Hướng dẫn giải Bài 18 (Trang 49 SGK Toán Đại số 9, Tập 2)
<p>Đưa các phương trình sau về dạng ax<sup>2</sup> + 2b'x + c = 0 và giải chúng. Sau đó, dùng bảng số hoặc máy tính để viết gần đúng nghiệm tìm được (làm tròn kết quả đến chữ số thập phân thứ hai):</p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">a</mi><mo>)</mo><mo> </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> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi mathvariant="normal">b</mi><mo>)</mo><mo> </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> </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> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi mathvariant="normal">d</mi><mo>)</mo><mo> </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> </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><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> </mo><mo>∆</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>≈</mo><mn>1</mn><mo>,</mo><mn>82</mn><mo>;</mo><mo> </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>≈</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> </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><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> </mo><mo>∆</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>≈</mo><mn>1</mn><mo>,</mo><mn>41</mn><mo>;</mo><mo> </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>≈</mo><mn>0</mn><mo>,</mo><mn>47</mn><mspace linebreak="newline"/><mspace linebreak="newline"/><mi mathvariant="normal">c</mi><mo>)</mo><mo> </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><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> </mo><mo>∆</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><</mo><mn>0</mn><mspace linebreak="newline"/><mi>Phương</mi><mo> </mo><mi>trình</mi><mo> </mo><mi>vô</mi><mo> </mo><mi>nghệm</mi><mo>.</mo><mspace linebreak="newline"/><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>)</mo><mo> </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><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> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>⇔</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> </mo><mi mathvariant="normal">b</mi><mo>'</mo><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>;</mo><mo> </mo><mo>∆</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> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </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>≈</mo><mn>4</mn><mo>,</mo><mn>56</mn><mo>;</mo><mo> </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>≈</mo><mn>0</mn><mo>,</mo><mn>44</mn></math></p>
<p>(Rõ ràng trong trường hợp này dùng công thức nghiệm thu gọn cũng không đơn giản hơn)</p>
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