Solve for x: (x - 0.16)x = 0.165.

Step-by-step solution

1. 1

   Expand and set the equation to zero. A product equal to a nonzero constant tells you nothing directly, so multiply out and collect everything on one side:

   $x^2-0.16x=0.165\qquad\Longrightarrow\qquad x^2-0.16x-0.165=0$

   Only now is the standard form $ax^2+bx+c=0$ available.

2. 2

   Read off the coefficients and check for easy factoring. Here $a=1$, $b=-0.16$, $c=-0.165$. With decimal coefficients there is no clean integer pair to guess, so the quadratic formula is the right tool.

3. 3

   Compute the discriminant.

   $\Delta=b^2-4ac=(-0.16)^2-4(1)(-0.165)=0.0256+0.66=0.6856$

   It is positive, so there are two distinct real roots. Note the sign care: $-4ac$ with $c$ negative adds $0.66$.

4. 4

   Take the square root and apply the formula.

   $\sqrt{0.6856}=0.8280097$

   $x=\frac{0.16\pm0.8280097}{2}$

5. 5

   Compute both roots.

   $x_1=\frac{0.16+0.8280097}{2}=\frac{0.9880097}{2}=0.4940048\approx0.4940$

   $x_2=\frac{0.16-0.8280097}{2}=\frac{-0.6680097}{2}=-0.3340048\approx-0.3340$

   One root is positive and one negative, which is expected since the product of the roots is $c/a=-0.165<0$.

6. 6

   Verify both roots in the original form. For $x_1$: $(0.4940048-0.16)(0.4940048)=0.3340048\times0.4940048=0.165000$ ✓. For $x_2$: $(-0.3340048-0.16)(-0.3340048)=(-0.4940048)(-0.3340048)=0.165000$ ✓. Also, the roots sum to $0.16=-b/a$ ✓.