If a and b are the solutions to the equation x^2 - 5x + 19 = 0, what is the value of (a - 1)(b - 1)?
+592 \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
\(x = {-(-5) \pm \sqrt{(-5)^2-4(1)(19)} \over 2(1)}\)
\(x = {5 \pm \sqrt{25-76} \over 2}\)
\(x = {5 \pm i\sqrt{51} \over 2}\)
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.)
