Final answer:
The coefficients a and b of the quadratic equation x^2 + ax + b = 0, with a and b as roots, are -1 and 1, respectively, based on the sum and product of roots for a quadratic equation.
Step-by-step explanation:
To find the coefficients a and b of the quadratic equation x² + ax + b = 0, given that a and b are also the roots of the equation, we can use the fact that for a quadratic equation of the form ax² + bx + c = 0, the sum of the roots (-b/a) equals the coefficient of x (which is 'a' in our case), and the product of the roots (c/a) equals the constant term 'b'.
Therefore, applying these relationships, we have:
Sum of roots (a + b) = -a
Product of roots (a * b) = b
Which leads to the system of equations:
a + b = -a
a * b = b
To solve for a, from the first equation a + a = -b, which simplifies to 2a = -b. To solve for b, we can see from the second equation that if ab = b and b is not zero, a must equal 1 (because if b = 0, then the original equation x² = 0 would not be quadratic). Hence, the coefficient a = -1, and coefficient b = 1, since anything multiplied by 1 remains unchanged.