Welcome to Vieta's formulas! These powerful mathematical relationships connect the coefficients of a polynomial equation directly to the properties of its roots. For a quadratic equation ax squared plus bx plus c equals zero with roots r one and r two, we have two key formulas: the sum of roots equals negative b over a, and the product of roots equals c over a.
Let's work through a concrete example. Consider the quadratic equation x squared minus 3x plus 2 equals zero. Here, a equals 1, b equals negative 3, and c equals 2. Using Vieta's formulas, the sum of roots equals negative b over a, which is negative negative 3 over 1, giving us 3. The product of roots equals c over a, which is 2 over 1, giving us 2. We can verify this by finding the actual roots: 1 and 2. Indeed, 1 plus 2 equals 3, and 1 times 2 equals 2.
Vieta's formulas extend to higher degree polynomials. For a cubic polynomial ax cubed plus bx squared plus cx plus d equals zero with roots r one, r two, and r three, we have three relationships. The sum of all roots equals negative b over a. The sum of products taken two at a time equals c over a. The product of all three roots equals negative d over a. Let's verify with the example x cubed minus 6x squared plus 11x minus 6 equals zero, which has roots 1, 2, and 3.