Final answer:
The quadratic function h(x) = -7x² + 10x - 5 has no real roots, as the discriminant, calculated from the coefficients of the function using the quadratic formula, is negative.
Step-by-step explanation:
To determine how many real roots the quadratic function h(x) = -7x² + 10x - 5 has, we can use the discriminant, which is part of the quadratic formula. The discriminant is given by b² - 4ac, where a, b, and c are the coefficients of x², x, and the constant term of the quadratic equation ax² + bx + c = 0, respectively. In this case, a = -7, b = 10, and c = -5. Calculating the discriminant, we get:
b² - 4ac = (10)² - 4(-7)(-5) = 100 - 140 = -40.
Since the discriminant is negative (-40), this means the quadratic equation has no real roots. Thus, the correct answer is C. 0.