What is the formula used to find the original quadratic equation given the roots?

The formula to find the original quadratic equation from its roots is derived from Vieta's formulas, which relate the coefficients of the polynomial to sums and products of its roots. For a quadratic equation in standard form ( ax^2 + bx + c = 0 ), if the roots are denoted as ( r_1 ) and ( r_2 ), Vieta's relations state that:

1. The sum of the roots ( r_1 + r_2 = -\frac{b}{a} ).

2. The product of the roots ( r_1 \cdot r_2 = \frac{c}{a} ).

When simplifying this for a monic quadratic equation (where ( a = 1 )), we can rewrite the equation as:

[

x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0.

]

Thus, the equation reflects that the quadratic's coefficients are determined by the sum and product of the roots.

Using this understanding, the correct formulation that reflects this relationship is ( X^2 - (sum of roots)x + (product of roots) = 0 ). This shows that