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# algebra

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If a and b are the solutions to the equation x^2 - 5x + 19 = 0, what is the value of (a - 1)(b - 1)?

Apr 5, 2021

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$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$

$$x = {-(-5) \pm \sqrt{(-5)^2-4(1)(19)} \over 2(1)}$$

$$x = {5 \pm \sqrt{25-76} \over 2}$$

$$x = {5 \pm i\sqrt{51} \over 2}$$

a = $$\frac{5 + i\sqrt{51}}{2}$$

b = $$\frac{5 - i\sqrt{51}}{2}$$

(a - 1)(b - 1)

$$(\frac{5+i\sqrt{51}}{2} - 1)(\frac{5-i\sqrt{51}}{2} - 1)$$

$$(\frac{3+i\sqrt{51}}{2} )(\frac{3-i\sqrt{51}}{2} )$$

$$\frac{(3+i\sqrt{51})(3-i\sqrt{51})}{4}$$

$$\frac{3^2-(i\sqrt{51})^2}{4}$$ ==> difference in squares

$$\frac{9 + 51}{4}$$

$$\frac{60}{4}$$

= 15

(edit: I realized I made a mistake. Hopefully this answer is correct.)

Apr 5, 2021
edited by Logarhythm  Apr 5, 2021
edited by Logarhythm  Apr 5, 2021