to determine the nature of the solutions use the discriminant
Given a quadratic equation in standard form : ax² + bx + c = 0
then the discriminant is b² - 4ac
• If b² - 4ac > 0 , then there are 2 real and distinct solutions
• If b² - 4ac = 0, there are 2 real and equal roots
• If b² - 4ac < 0 , there are no real solutions
for x² - 3x + 8 = 0 with a = 1, b= - 3 and c = 8
b² - 4ac = (- 3)² - (4 × 1 × 8 ) = 9 - 32 = - 23
Hence there are no real solutions to the equation
