Solve 0.097x^2 - 2.20765x + 4.37114 = 0.

Step-by-step solution

1. 1

   Set up the quadratic formula. With $a=0.097,\qquad b=-2.20765,\qquad c=4.37114,$ there is no integer factorisation available, so use $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.$ Note that $a$ is small, which is a warning sign: the final division by $2a=0.194$ multiplies every rounding error by about $5$.

2. 2

   Evaluate the discriminant exactly as written. $b^{2}=(-2.20765)^{2}=4.8737185225,$ $4ac=4(0.097)(4.37114)=1.6960023200,$ $D=4.8737185225-1.6960023200=3.1777162025.$ Keeping all ten decimals here is what makes the roots trustworthy to six figures later.

3. 3

   Take the square root and read off the root count. $\sqrt{D}=\sqrt{3.1777162025}=1.78261499\ldots$ Because $D>0$, there are two distinct real roots; the parabola opens upward and crosses the axis twice.

4. 4

   Divide by $2a$ to finish. $x=\frac{2.20765\pm 1.78261499}{0.194}.$ The plus branch gives $x=\frac{3.99026499}{0.194}=20.56837624,$ and the minus branch gives $x=\frac{0.42503501}{0.194}=2.19090211.$

5. 5

   Verify with the sum and product of the roots. Vieta's relations say the roots must sum to $-b/a=2.20765/0.097=22.7592784$ and multiply to $c/a=4.37114/0.097=45.0632990$. Our values give $20.56837624+2.19090211=22.75927835$ and $20.56837624\times 2.19090211=45.06329897$, so both roots are correct.