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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

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