Solve for x: (x - 0.16)x = 16.5.

Step-by-step solution

1. 1

   Expand and put it in standard form. A product equal to a nonzero number cannot be attacked by the zero-product rule, so multiply out and collect:

   $x^2-0.16x=16.5\qquad\Longrightarrow\qquad x^2-0.16x-16.5=0$

2. 2

   Identify the coefficients and choose the method. $a=1$, $b=-0.16$, $c=-16.5$. The decimals rule out easy factoring, so use the quadratic formula.

3. 3

   Compute the discriminant carefully.

   $\Delta=b^2-4ac=(-0.16)^2-4(1)(-16.5)=0.0256+66=66.0256$

   The two minus signs in $-4ac$ with $c<0$ combine to add $66$ — a frequent sign slip. Since $\Delta>0$ there are two distinct real roots.

4. 4

   Take the square root and set up both roots.

   $\sqrt{66.0256}=8.12561$

   $x=\frac{0.16\pm8.12561}{2}$

   Note $\sqrt{66.0256}$ is only barely above $\sqrt{66}=8.1240$ — the $0.0256$ contributes almost nothing.

5. 5

   Evaluate both roots.

   $x_1=\frac{0.16+8.1256138}{2}=\frac{8.2856138}{2}=4.1428069\approx4.143$

   $x_2=\frac{0.16-8.1256138}{2}=\frac{-7.9656138}{2}=-3.9828069\approx-3.983$

   One positive, one negative — as expected, since the product of the roots is $c/a=-16.5<0$.

6. 6

   Verify both in the original equation. For $x_1$: $(4.1428069-0.16)(4.1428069)=3.9828069\times4.1428069=16.4999999$ ✓. For $x_2$: $(-3.9828069-0.16)(-3.9828069)=(-4.1428069)(-3.9828069)=16.4999999$ ✓. The roots also sum to $0.16=-b/a$ ✓.