The give equation has two complex number solutions.
Work Step by Step
Subtract $7$ to both sides to obtain: $$-8x^2+10x-7=7-7 \\-8x^2+10x-7=0$$ This this equation has $a=-8$, $b=10$, and $c=-7$. RECALL: (1) The discriminant is equal to $b^2-4ac$. (2) A quadratic equation has the following types of solutions based on the value of the discriminant: (a) when $b^2-4ac\gt0$, the equation has two unequal rational solutions; (b) when $b^2-4ac=0$, the equation has one, repeated rational solution; and (c) when $b^2-4ac\lt0$, the equation has two complex number solutions; The discriminant of the equation above is: $$b^2-4ac = 10^2 - 4(-8)(-7) = 100-224=-124$$ The discriminant is negative. Thus, the give equation has two complex number solutions.