Solve -0.2 = 4.3957x2 + 1.7334x + 0.0496.

Step-by-step solution

1. 1

   Move everything to one side. A quadratic can only be attacked once it reads $ax^2+bx+c=0$. Adding $0.2$ to both sides gives $4.3957x^2 + 1.7334x + 0.0496 + 0.2 = 0$, i.e. $4.3957x^2 + 1.7334x + 0.2496 = 0$, so $a = 4.3957$, $b = 1.7334$, $c = 0.2496$.

2. 2

   Decide to use the quadratic formula. The coefficients are four-decimal experimental numbers, so there is no factor pair to guess and completing the square would only move the ugly arithmetic around. The formula $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$ handles any coefficients.

3. 3

   Compute the discriminant carefully. $b^2 = 1.7334^2 = 3.00467556$ and $4ac = 4(4.3957)(0.2496) = 4.38866688$. Therefore $\Delta = b^2 - 4ac = 3.00467556 - 4.38866688 = -1.38399132.$ Both products must be carried to full precision here, because $\Delta$ is a difference of two nearly equal numbers and rounding early can change its sign.

4. 4

   Read the sign of the discriminant. $\Delta < 0$, so the parabola $y = 4.3957x^2 + 1.7334x + 0.2496$ opens upward yet never touches the $x$-axis. The equation has no real solutions - and since $\Delta<0$ the original statement $-0.2 = 4.3957x^2+1.7334x+0.0496$ is false for every real $x$.

5. 5

   Write the complex pair. Using $\sqrt{-1.38399132} = i\sqrt{1.38399132}$ and $\sqrt{1.38399132} \approx 1.1764316$, with $2a = 8.7914$: $x = \frac{-1.7334 \pm 1.1764316\,i}{8.7914} \approx -0.19717 \pm 0.13382\,i.$

6. 6

   Sanity-check with the vertex. The two complex roots must share the real part $-\dfrac{b}{2a} = -\dfrac{1.7334}{8.7914} = -0.19717$, which is exactly the $x$-coordinate of the vertex, and the minimum value $c - \dfrac{b^2}{4a} = 0.2496 - 0.07873 = 0.07873 > 0$ confirms the parabola stays above the axis.