Editing Quadratic equation (section)

====Vieta's formulas====
{{Main|Vieta's formulas}}

''Vieta's formulas'' (named after [[François Viète]]) are the relations
<math display="block"> x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}</math> 
between the roots of a quadratic polynomial and its coefficients. They result from comparing term by term the relation
<math display="block">\left( x - x_1 \right) \left( x-x_2 \right ) = x^2 - \left( x_1+x_2 \right)x +x_1 x_2 = 0</math>
with the equation
<math display="block"> x^2 + \frac ba x +\frac ca = 0.</math>

The first Vieta's formula is useful for graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the [[Quadratic function#Vertex|vertex]],  the vertex's {{math|''x''}}-coordinate is located at the average of the roots (or intercepts). Thus the {{math|''x''}}-coordinate of the vertex is
<math display="block"> x_V = \frac {x_1 + x_2} {2} = -\frac{b}{2a}.</math>
The {{math|''y''}}-coordinate can be obtained by substituting the above result into the given quadratic equation, giving
<math display="block"> y_V = - \frac{b^2}{4a} + c = - \frac{ b^2 - 4ac} {4a}.</math>
Also, these formulas for the vertex can be deduced directly from the formula (see [[Completing the square]])
<math display="block">ax^2+bx+c=a \left(x+\frac b{2a}\right)^2 - \frac{b^2-4ac}{4a}.</math>

For numerical computation, Vieta's formulas provide a useful method for finding the roots of a quadratic equation in the case where one root is much smaller than the other. If {{math|{{!}}''x''<sub>2</sub>{{!}} &lt;&lt; {{!}}''x''<sub>1</sub>{{!}}}}, then {{math|''x''<sub>1</sub> + ''x''<sub>2</sub> ≈ ''x''<sub>1</sub>}}, and we have the estimate:
<math display="block"> x_1 \approx -\frac{b}{a} .</math>
The second Vieta's formula then provides:
<math display="block">x_2 = \frac{c}{a x_1} \approx -\frac{c}{b} .</math>
These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large {{math|''b''}}), which causes [[round-off error]] in a numerical evaluation. The figure shows the difference between{{clarify|reason=without indication on the numerical accuracy, the figure and its discussion are nonsensical. At least the difference with the exact value of the root must also appear.|date=September 2021}} (i)&nbsp;a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii)&nbsp;an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient {{math|''b''}} increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as {{math|''b''}} increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse.

This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see [[Step response]]).