For the quadratic z^2 - z + 2 = 0, which statement about its roots is true?

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Multiple Choice

For the quadratic z^2 - z + 2 = 0, which statement about its roots is true?

For a quadratic, the sum and product of its roots relate directly to its coefficients: if z1 and z2 are the roots of ax^2 + bx + c = 0, then z1 + z2 = -b/a and z1 z2 = c/a. Here the equation is z^2 - z + 2 = 0, so a = 1, b = -1, c = 2. The sum of the roots is -(-1)/1 = 1, and the product is 2/1 = 2. So the roots have product 2 and sum 1. The discriminant is negative (1 - 8 = -7), meaning the roots are complex, but these sum and product values still hold. Expanding (z - z1)(z - z2) gives z^2 - (z1 + z2) z + z1 z2, which becomes z^2 - 1 z + 2, matching the original equation.

For a quadratic, the sum and product of its roots relate directly to its coefficients: if z1 and z2 are the roots of ax^2 + bx + c = 0, then z1 + z2 = -b/a and z1 z2 = c/a. Here the equation is z^2 - z + 2 = 0, so a = 1, b = -1, c = 2. The sum of the roots is -(-1)/1 = 1, and the product is 2/1 = 2. So the roots have product 2 and sum 1. The discriminant is negative (1 - 8 = -7), meaning the roots are complex, but these sum and product values still hold. Expanding (z - z1)(z - z2) gives z^2 - (z1 + z2) z + z1 z2, which becomes z^2 - 1 z + 2, matching the original equation.

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