Solve -4.4912x^2 - 0.5182x + 30.066 = 0 for x.

Step-by-step solution

1. 1

   Factor the leading coefficient out of the first two terms.

   $-4.4912\left(x^2+\frac{0.5182}{4.4912}x\right)+30.066=-4.4912\left(x^2+0.115381x\right)+30.066$

   Only the $x^2$ and $x$ terms go inside the bracket; the constant stays outside, which is what keeps the bookkeeping straight.

2. 2

   Complete the square inside the bracket. Half the linear coefficient is $0.0576905$, and its square is $0.00332820$:

   $x^2+0.115381x=\left(x+0.0576905\right)^2-0.00332820$

3. 3

   Distribute the leading coefficient back.

   $-4.4912\left(x+0.0576905\right)^2+4.4912(0.00332820)+30.066=-4.4912\left(x+0.0576905\right)^2+30.08095$

   This is vertex form: the parabola peaks at $\left(-0.05769,\,30.08095\right)$. Because the maximum is positive and the parabola opens downward, two real roots must exist.

4. 4

   Solve the vertex form for $x$.

   $\left(x+0.0576905\right)^2=\frac{30.08095}{4.4912}=6.697710\;\Longrightarrow\;x+0.0576905=\pm 2.588002$

   $x=-0.0576905\pm 2.588002$

5. 5

   Evaluate and check.

   $x_1=-2.64569,\qquad x_2=2.53031$

   Vieta confirms both: the sum should equal $-\tfrac{b}{a}=-\tfrac{-0.5182}{-4.4912}=-0.11538$, and $-2.64569+2.53031=-0.11538\;\checkmark$; the product should equal $\tfrac{c}{a}=\tfrac{30.066}{-4.4912}=-6.69420$, and $(-2.64569)(2.53031)=-6.69423\;\checkmark$. The roots are symmetric about the vertex abscissa $-0.05769$, exactly as vertex form predicts.