What This Calculator Does
This tool applies Vieta formulas to a quadratic equation written in standard form \(ax^{2} + bx + c = 0\). Without ever solving for the actual roots, it instantly returns the sum of the roots and the product of the roots. This is incredibly handy for checking your work, building equations from known roots, or solving problems where only the sum and product matter.
How to Use It
Enter the three coefficients: a (the coefficient of \(x^{2}\)), b (the coefficient of \(x\)), and c (the constant term). The coefficient a must not be zero, otherwise the equation is linear rather than quadratic. The calculator returns the sum \(\left(-\frac{b}{a}\right)\), the product \(\left(\frac{c}{a}\right)\), and the discriminant \(\left(b^{2} - 4ac\right)\), which tells you the nature of the roots.
The Formula Explained
For any quadratic \(ax^{2} + bx + c = 0\) with roots \(x_{1}\) and \(x_{2}\), Vieta formulas state that $$x_{1} + x_{2} = -\frac{b}{a} \qquad x_{1} \cdot x_{2} = \frac{c}{a}$$ These follow directly from factoring: $$a(x - x_{1})(x - x_{2}) = ax^{2} - a(x_{1}+x_{2})x + a\cdot x_{1}x_{2}$$ then matching coefficients with the original equation.
Worked Example
Take \(x^{2} - 5x + 6 = 0\), so \(a = 1\), \(b = -5\), \(c = 6\). The sum of roots is $$-\frac{b}{a} = -\frac{-5}{1} = 5$$ and the product is $$\frac{c}{a} = \frac{6}{1} = 6$$ Indeed the roots are 2 and 3, which add to 5 and multiply to 6. The discriminant is $$(-5)^{2} - 4(1)(6) = 25 - 24 = 1$$ a positive perfect square confirming two distinct rational roots.
FAQ
Do I need to solve the quadratic first? No. Vieta formulas give the sum and product directly from the coefficients, no solving required.
What does the discriminant tell me? If \(b^{2} - 4ac\) is positive there are two real roots, if zero there is one repeated root, and if negative the roots are complex conjugates.
Can a be zero? No. If \(a = 0\) the equation is linear and Vieta formulas for a quadratic do not apply.
