Final answer:
There is one quadratic polynomial with real coefficients where the set of roots equals the set of coefficients, which is the polynomial x^2 - x.
Step-by-step explanation:
The question is asking how many quadratic polynomials with real coefficients exist such that the roots of the polynomial are exactly the same as its coefficients. To solve this, let's consider a quadratic equation in the standard form:
ax^2 + bx + c = 0
For the roots of this equation to be the same as the coefficients, we must have a situation where a and b are the roots of the polynomial. This means:
x = a and x = b
Using the quadratic formula, x = (-b ± sqrt(b^2 - 4ac)) / (2a), let's find the conditions for this to occur. For simplicity, let's restrict our attention to non-complex roots (real roots), and therefore we need the discriminant, b^2 - 4ac, to be non-negative.
By the nature of the quadratic formula, it is impossible to have two distinct real roots equal to the coefficients since one of the roots will involve the term -b, leading to a contradiction. Therefore, we must look for cases where both roots are the same (a double root), which happens when the discriminant is zero.
Setting the discriminant to zero gives us the condition: b^2 = 4ac, and as both roots are the same, we must have -b/2a = a = b, leading to a zero for coefficient c. This implies that the only solution to the problem is the trivial case where the quadratic polynomial is x^2 - x, which has a double root at x = 1. Hence, there is one such quadratic polynomial.