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x2 x 1 2 This is in the form fraction fraction so we will 5 cross multiply 2x 8 5 x 1 2 x2 2x 8 5x 5 2x2 4x 16 2x2 4x 16 2x2 4x 16 2x2 x 21 0 2x 7 x 3 0 2x 7 0 7 7 2x 7 x 7 2 x 3 0 3 3 x 3
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4 x 1 4 x x 2 x 2x First factor each denominator, second nd the LCD, and third multiply all three terms by the LCD
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4 x 1 4 x x 2 x x 2
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x x 2 4 x x 2 x 1 x x 2 4 1 x x 2 1 x 1 x 2 4 x 2 x 1 4x 4 x2 x 2 4x x2 x 2 4x 4x 4x x2 3x 2 0 x 2 x 1 0 x 2 0 2 2 x 2 3 6x 24 2 x 4 x 2x 8 x 1 0 1 1 x 1 Cross multiply
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3 x2 2x 8 x 4 6x 24 3x2 6x 24 6x2 48x 96 6x2 48x 96 6x2 48x 96 9x2 54x 72 0 1 1 9x2 54x 72 0 9 9 2 x 6x 8 0 x 4 x 2 0 x 4 0 4 4 x 4 x 2 0 2 2 x 2
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We cannot let x be 4 because x 4 leads to a zero in the denominator of 3 : The only solution is x 2: x 4
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Practice
Because all of these problems factor, factoring is used in the solutions If factoring takes too long on some of these, you may use the quadratic formula 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 3x 3x x 4 2 x 1 6 2x 3 x 2 x 2 2x 1 12x 3 x 3 x2 9 x 3 2x 1 3x 8x 7 2 x 1 x 2 x x 2 4x 1 x 5 24 x 1 1 x x 2x 3 2 2 x 1 x x x 2x 3 3x 4 2x 1 x 3x 2 1 8 x 5 3x 3x 15 1 2 3 x 3 2 x 4 x 1 x 3 x 3x 4 4 3 1 6 x 1 x 1 2x x2 1
Solutions
1: 3x 3x x 4 2
Quadratic Equations
3x 2 x 4 3x 6x 3x2 12x 3x2 12x 3x2 12x 3x2 6x 0 3x x 2 0 3x 0 0 x 3 x 0 2: x 1 6 2x 3 x 2 x 1 x 2 6 2x 3 x2 3x 2 12x 18 12x 18 12x 18 x2 15x 16 0 x 16 x 1 0 x 16 0 16 16 x 16 3: x 2 2x 1 12x 3 x 3 x2 9 x 3 Denominator factored: LCD x 3 x 3 x 3 x 3 x 2 2x 1 x 3 x 3 x 3 x 3 x 3 12x 3 x 3 x 2 2x 1 12x 3 x 3 x 3 x 3 x 3 x 1 0 1 1 x 1 x 2 0 2 2 x 2
x 3 x 3
Quadratic Equations
x 3 x 2 2x 1 x 3 12x 3 x2 5x 6 2x 1 12x2 33x 9 x2 7x 7 12x2 33x 9 x2 7x 7 x2 7x 7 0 11x2 40x 16 0 11x 4 x 4 11x 4 0 4 4 11x 4 4 x 11 4: x 4 0 4 4 x 4
2x 1 3x 8x 7 2 x 1 x 2 x x 2 2x 1 3x 8x 7 Denominator factored: x 1 x 2 x 2 x 1 LCD x 1 x 2 x 1 x 2 2x 1 3x x 1 x 2 x 1 x 2 8x 7 x 1 x 2 x 2 x 1
x 2 2x 1 3x x 1 8x 7 2x2 5x 2 3x2 3x 8x 7 x2 8x 2 8x 7 x2 8x 2 x2 8x 2 0 x2 9 0 x 3 x 3 x 3 0 3 3 x 3 x 3 0 3 3 x 3
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