Hướng dẫn giải Bài 33 (Trang 23 SGK Toán Đại số 8, Tập 2)

Tìm các giá trị của a sao cho các biểu thức sau có giá trị bằng 2: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mfrac><mrow><mn>3</mn><mi>a</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>3</mn><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mn>3</mn></mrow><mrow><mi>a</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><mfrac><mn>10</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mrow><mn>3</mn><mi>a</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>4</mn><mi>a</mi><mo>+</mo><mn>12</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>7</mn><mi>a</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>6</mn><mi>a</mi><mo>+</mo><mn>18</mn></mrow></mfrac></math> Giải a) Bài toán tương đương với tìm các giá trị của a sao cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>a</mi><mo>+</mo><mn>1</mn><mo>≠</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>+</mo><mn>3</mn><mo>≠</mo><mn>0</mn></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>a</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>3</mn><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mn>3</mn></mrow><mrow><mi>a</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mo> </mo><mo>(</mo><mn>1</mn><mo>)</mo></math> Điều này có nghĩa là phải giải phương trình (1) với ẩn số a ĐKXĐ: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≠</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mi>a</mi><mo>≠</mo><mo>-</mo><mn>3</mn></math> Quy đồng và khử mẫu <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>3</mn><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>a</mi><mo>+</mo><mn>3</mn><mo>)</mo><mo>+</mo><mo>(</mo><mi>a</mi><mo>-</mo><mn>3</mn><mo>)</mo><mo>(</mo><mn>3</mn><mi>a</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>2</mn><mo>(</mo><mi>a</mi><mo>+</mo><mn>3</mn><mo>)</mo><mo>(</mo><mn>3</mn><mi>a</mi><mo>+</mo><mn>1</mn><mo>)</mo><mspace linebreak="newline"/><mo>⇔</mo><mn>6</mn><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mo>=</mo><mn>6</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>20</mn><mi>a</mi><mo>+</mo><mn>6</mn><mo>⇔</mo><mn>20</mn><mi>a</mi><mo>=</mo><mo>-</mo><mn>12</mn><mo>⇔</mo><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>5</mn></mfrac><mo> </mo><mo>(</mo><mi>n</mi><mi>h</mi><mi>a</mi><mi>n</mi><mo>)</mo></math> Kết luận: giá trị cần tìm của a là <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>5</mn></mfrac></math> b) Ta cần giải phương trình: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>10</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mrow><mn>3</mn><mi>a</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>4</mn><mi>a</mi><mo>+</mo><mn>12</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>7</mn><mi>a</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>6</mn><mi>a</mi><mo>+</mo><mn>18</mn></mrow></mfrac><mo>=</mo><mn>2</mn><mo> </mo><mo>(</mo><mn>2</mn><mo>)</mo><mspace linebreak="newline"/><mi>Đ</mi><mi>K</mi><mi>X</mi><mi>Đ</mi><mo>:</mo><mo> </mo><mi>a</mi><mo>≠</mo><mo>-</mo><mn>3</mn></math> Quy đồng và khử mẫu <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo>(</mo><mn>2</mn><mi>a</mi><mo>+</mo><mn>6</mn><mo>)</mo><mo>-</mo><mn>3</mn><mo>(</mo><mn>3</mn><mi>a</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>-</mo><mn>2</mn><mo>(</mo><mn>7</mn><mi>a</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>=</mo><mn>12</mn><mo>(</mo><mn>2</mn><mi>a</mi><mo>+</mo><mn>6</mn><mo>)</mo><mspace linebreak="newline"/><mo>⇔</mo><mn>7</mn><mi>a</mi><mo>=</mo><mn>47</mn><mo>⇔</mo><mi>a</mi><mo>=</mo><mfrac><mn>47</mn><mn>7</mn></mfrac></math> Kết luận: Giá trị cần tìm là <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>47</mn><mn>7</mn></mfrac></math>