Question

Solve each of the following equation by completing the square or quadratic formula

1. 2x² - 5x - 3 = 0
2. x² + 4x -7 = 0
3. 2x² - 4x = -1


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Answer 1

Answer:

$1)2 {x}^{2} - 5x - 3 = 0 \\ a = 2 ,\: b = ( - 5), \: c = ( - 3) \\ x = \frac{( - b)± \sqrt{ {b}^{2} - 4ac } }{2a} \\ = \frac{ - ( - 5)± \sqrt{ {( - 5)}^{2} - 4(2)( - 3) } }{2(2)} \\ = \frac{5± \sqrt{25 + 25} }{4} \\ = \frac{5± \sqrt{50} }{4} \\ = \frac{5±5 \sqrt{2} }{4} \\ = \frac{5 + 5 \sqrt{2} }{4} \: and \: \frac{5 - 5 \sqrt{2} }{4} \\ = \frac{5(1 + \sqrt{2} )}{4} \: and \: \frac{5(1 - \sqrt{2}) }{4} \\ 2) {x}^{2} + 4x - 7 = 0 \\ a = 1 ,\: b = 4, \: c = ( - 7) \\ x = \frac{( - b)± \sqrt{ {b}^{2} - 4ac } }{2a} \\ = \frac{( - 4)± \sqrt{ {( - 4)}^{2} - 4(1)( - 7) } }{2(1)} \\ = \frac{( - 4)± \sqrt{16 + 28} }{2} \\ = \frac{( - 4)± \sqrt{44} }{2} \\ = \frac{( - 4)±2\sqrt{11} }{4} \\ = \frac{2( - 2± \sqrt{11} )} {4} \\ = \frac{( - 2)± \sqrt{11} }{2} \\ = \frac{( - 2) + \sqrt{11} }{2} \: and \: \frac{( - 2) - \sqrt{11} }{2} \\ 3)2 {x}^{2} - 4x = ( - 1) \\ 2 {x}^{2} - 4x + 1 = 0 \\ a = 2 ,\: b = ( - 4), \: c = 1 \\ x = \frac{( - b)± \sqrt{ {b}^{2} - 4ac} }{2a} \\ = \frac{ - ( - 4)± \sqrt{ {( - 4)}^{2} - 4(2)(1) } }{2(2)} \\ = \frac{4± \sqrt{16 - 8} }{4} \\ = \frac{4± \sqrt{8} }{4} \\ = \frac{4±2 \sqrt{2} }{4} \\ = \frac{2(2± \sqrt{2}) }{4} \\ = \frac{2± \sqrt{2} }{2} \\ = \frac{2 + \sqrt{2} }{2} \: and \: \frac{2 + \sqrt{2} }{2}$