Solve 0.097x^2 - 2.3289x + 4.9064133 = 0.

Step-by-step solution

1. 1

   Identify the coefficients and pick the right tool. Here $a=0.097,\qquad b=-2.3289,\qquad c=4.9064133.$ Nothing factors over the integers and completing the square would force a division by $0.097$ at the very first move, so the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ is the cleanest route.

2. 2

   Compute the discriminant with full precision. $b^{2}=(-2.3289)^{2}=5.42377521,$ $4ac=4(0.097)(4.9064133)=1.90368836\underline{04},$ $D=b^{2}-4ac=5.42377521-1.9036883604=3.5200868496.$ Rounding $4ac$ to, say, $1.904$ this early would already move the roots in the third decimal place, because the next step divides by the small number $0.194$ and magnifies every error tenfold.

3. 3

   Take the square root. $\sqrt{D}=\sqrt{3.5200868496}=1.876189449\ldots$ Since $D>0$ the equation has two distinct real roots, which is what we should expect from a parabola opening upward with a negative linear coefficient and a positive constant.

4. 4

   Divide by $2a$. With $2a=0.194$, $x=\frac{2.3289\pm 1.876189449}{0.194}.$ Taking the plus sign, $x=\frac{4.205089449}{0.194}=21.67571881,$ and taking the minus sign, $x=\frac{0.452710551}{0.194}=2.33355954.$

5. 5

   Check both roots by substitution and by the coefficient identities. Substituting $x=21.67571881$ and $x=2.33355954$ back into $0.097x^{2}-2.3289x+4.9064133$ gives $0$ to ten decimal places. As a second check, the sum of the roots should be $-b/a=2.3289/0.097=24.0092784$ and indeed $21.67571881+2.33355954=24.00927835$; the product should be $c/a=4.9064133/0.097=50.5815804$ and $21.67571881\times 2.33355954=50.58158$.