Solve -4.6173x^2 - 0.2906x + 20.029 = 0 for x.

Step-by-step solution

1. 1

   Clear the negative leading coefficient. Multiplying an equation by $-1$ changes no roots but removes the most error-prone sign:

   $-4.6173x^2-0.2906x+20.029=0\iff 4.6173x^2+0.2906x-20.029=0$

   Now $a=4.6173$, $b=0.2906$, $c=-20.029$, and the denominator $2a$ in the quadratic formula is positive.

2. 2

   Compute the discriminant.

   $\Delta=b^2-4ac=0.084448-4(4.6173)(-20.029)=0.084448+369.9196=370.00406$

   $\sqrt{\Delta}=19.23549$

   The negative $c$ guarantees $-4ac>0$, so two real roots of opposite signs are certain before any further work.

3. 3

   Apply the quadratic formula.

   $x=\frac{-0.2906\pm 19.23549}{2(4.6173)}=\frac{-0.2906\pm 19.23549}{9.2346}$

4. 4

   Evaluate both roots.

   $x_1=\frac{-0.2906-19.23549}{9.2346}=-2.11445,\qquad x_2=\frac{-0.2906+19.23549}{9.2346}=2.05151$

   With the positive leading coefficient the plus branch now gives the larger root, as intuition expects.

5. 5

   Verify by substitution and by Vieta. Putting $x=2.05151$ into the original: $-4.6173(4.20869)-0.2906(2.05151)+20.029=-19.43278-0.5962+20.029\approx 0\;\checkmark$. Vieta: the sum should be $-\tfrac{0.2906}{4.6173}=-0.06294$ and $-2.11445+2.05151=-0.06294\;\checkmark$; the product should be $\tfrac{-20.029}{4.6173}=-4.33782$ and $(-2.11445)(2.05151)=-4.33782\;\checkmark$.