Solve 0.097x^2 - 2.3289x + 12.4464133 = 0.

Step-by-step solution

1. 1

   Identify the coefficients and expect sensitivity. Here $a=0.097$, $b=-2.3289$, $c=12.4464133$. The leading coefficient is small, so $2a=0.194$ is small too and the final division magnifies every earlier error about fivefold. That makes the discriminant computation the critical step.

2. 2

   Compute $b^{2}$ and $4ac$ separately and carefully.

   $b^{2}=(-2.3289)^{2}=5.42377521$

   $4ac=4\times 0.097\times 12.4464133=4\times 1.2073020901=4.8292083604$

   Compute $ac$ first, then multiply by 4; doing all three at once is where a digit gets dropped.

3. 3

   Form the discriminant.

   $D=5.42377521-4.8292083604=0.5945668496$

   It is positive but small, so the two roots will be relatively close together. (Mistyping $4ac$ as $4.8296083604$ — one digit — would give $D=0.5941668496$ and shift both roots by about $0.002$.)

4. 4

   Take the square root.

   $\sqrt{D}=\sqrt{0.5945668496}=0.7710816102$

5. 5

   Apply the quadratic formula.

   $x=\frac{2.3289\pm 0.7710816102}{0.194}$

   $x_{1}=\frac{3.0999816102}{0.194}=15.979287,\qquad x_{2}=\frac{1.5578183898}{0.194}=8.029992$

6. 6

   Check with Vieta relations. The sum of the roots must be $-\dfrac{b}{a}=\dfrac{2.3289}{0.097}=24.009278$, and indeed $15.979287+8.029992=24.009279$ ✓. The product must be $\dfrac{c}{a}=\dfrac{12.4464133}{0.097}=128.31354$, and $15.979287\times 8.029992=128.31354$ ✓. Two independent checks confirm both roots.