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108. DISCUSS: The Cubic Formula The Quadratic Formula can be used to solve any quadratic (or second-degree) equation. You might have wondered whether similar formulas exist for cubic (third-degree), quartic (fourth-degree), and higher- degree equations. For the depressed cubic x* + px +q = 0. Cardano (page 328) found the following formula for one solution: b. x = 27 2 V 4 27 A formula for quartic equations was discovered by the Ital- ian mathematician Ferrari in 1540. In 1824 the Norwegian mathematician Niels Henrik Abel proved that it is impossi- ble to write a quintic formula, that is, a formula for fifth- degree equations. Finally, Galois (page 313) gave a criterion for determining which equations can be solved by a formula involving radicals. Use the formula given above to find a solution for the fol- lowing equations. Then solve the equations using the meth- ods you learned in this section. Which method is easier? (a) x - 3x + 2 = 0 (b) x - 27x - 54 = 0 (c) x +3x + 4 = 0

108. DISCUSS: The Cubic Formula The Quadratic Formula can be used to solve any quadratic (or second-degree) equation. You might have wondered whether similar formulas exist for cubic (third-degree), quartic (fourth-degree), and higher- degree equations. For the depressed cubic x* + px +q = 0. Cardano (page 328) found the following formula for one solution: b. x = 27 2 V 4 27 A formula for quartic equations was discovered by the Ital- ian mathematician Ferrari in 1540. In 1824 the Norwegian mathematician Niels Henrik Abel proved that it is impossi- ble to write a quintic formula, that is, a formula for fifth- degree equations. Finally, Galois (page 313) gave a criterion for determining which equations can be solved by a formula involving radicals. Use the formula given above to find a solution for the fol- lowing equations. Then solve the equations using the meth- ods you learned in this section. Which method is easier? (a) x - 3x + 2 = 0 (b) x - 27x - 54 = 0 (c) x +3x + 4 = 0

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter3: Polynomial And Rational Functions
Section3.4: Real Zeros Of Polynomials
Problem 108E
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108. DISCUSS: The Cubic Formula The Quadratic Formula can be used to solve any quadratic (or second-degree) equation. You might have wondered whether similar formulas exist for cubic (third-degree), quartic (fourth-degree), and higher- degree equations. For the depressed cubic x* + px +q = 0. Cardano (page 328) found the following formula for one solution: b. x = 27 2 V 4 27 A formula for quartic equations was discovered by the Ital- ian mathematician Ferrari in 1540. In 1824 the Norwegian mathematician Niels Henrik Abel proved that it is impossi- ble to write a quintic formula, that is, a formula for fifth- degree equations. Finally, Galois (page 313) gave a criterion for determining which equations can be solved by a formula involving radicals. Use the formula given above to find a solution for the fol- lowing equations. Then solve the equations using the meth- ods you learned in this section. Which method is easier? (a) x - 3x + 2 = 0 (b) x - 27x - 54 = 0 (c) x +3x + 4 = 0
Transcribed Image Text:108. DISCUSS: The Cubic Formula The Quadratic Formula can be used to solve any quadratic (or second-degree) equation. You might have wondered whether similar formulas exist for cubic (third-degree), quartic (fourth-degree), and higher- degree equations. For the depressed cubic x* + px +q = 0. Cardano (page 328) found the following formula for one solution: b. x = 27 2 V 4 27 A formula for quartic equations was discovered by the Ital- ian mathematician Ferrari in 1540. In 1824 the Norwegian mathematician Niels Henrik Abel proved that it is impossi- ble to write a quintic formula, that is, a formula for fifth- degree equations. Finally, Galois (page 313) gave a criterion for determining which equations can be solved by a formula involving radicals. Use the formula given above to find a solution for the fol- lowing equations. Then solve the equations using the meth- ods you learned in this section. Which method is easier? (a) x - 3x + 2 = 0 (b) x - 27x - 54 = 0 (c) x +3x + 4 = 0
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