Solve -4.3289x^2 - 0.9671x + 40.328 = 0 for x.

Step-by-step solution

1. 1

   Read off the coefficients and predict the shape.

   $a=-4.3289,\qquad b=-0.9671,\qquad c=40.328$

   Since $a<0$ and $c>0$, the downward parabola crosses the axis on both sides of the origin, so expect one negative and one positive root.

2. 2

   Compute the discriminant.

   $\Delta=(-0.9671)^2-4(-4.3289)(40.328)=0.935282+698.303517=699.238799$

   $\sqrt{\Delta}=26.443124$

3. 3

   Apply the quadratic formula.

   $x=\frac{0.9671\pm 26.443124}{2(-4.3289)}=\frac{0.9671\pm 26.443124}{-8.6578}$

   $x_1=\frac{0.9671+26.443124}{-8.6578}=-3.16596,\qquad x_2=\frac{0.9671-26.443124}{-8.6578}=2.94255$

4. 4

   Verify with the sum of the roots. Vieta’s first formula says $x_1+x_2=-\tfrac{b}{a}$:

   $-\frac{-0.9671}{-4.3289}=-0.223405,\qquad -3.16596+2.94255=-0.22341\;\checkmark$

5. 5

   Verify with the product of the roots. Vieta’s second formula says $x_1x_2=\tfrac{c}{a}$:

   $\frac{40.328}{-4.3289}=-9.3159925,\qquad (-3.16596)(2.94255)=-9.3159956$

   The two already agree to six significant figures; the difference in the seventh is nothing but the rounding of the roots to five decimals, and carrying more digits ($-3.1659572$ and $2.9425517$) tightens the product to $-9.3159927$ $\checkmark$. Together the two Vieta checks pin down both roots without re-running the formula.