Chapter 5.5, Problem 51PPS

To determine

To find: the solution of the equation and round off to nearest 100^th if necessary.

Answer to Problem 51PPS

  $\text{x=-1}\text{.26375}$ and $\text{x=0}\text{.26375}$ .

Explanation of Solution

Given:

  ${\text{x}}^{\text{2}}\text{+x+}\frac{\text{1}}{\text{3}}\text{=}\frac{\text{2}}{\text{3}}$ .

Concept used:

To solve the ${\text{ax}}^{\text{2}}\text{+bx+c=0}$ by completing the square:

1. Transform the equation so that the constant term $\text{c}$, is alone on the right side.
2. If $\text{a}$ ,the leading coefficient of ${\text{x}}^2$ term is not equal to $1$ , divide both side by $\text{a}$ .
3. Add the square of half the coefficient of the x term, ${\left(\frac{\text{b}}{\text{2a}}\right)}^{\text{2}}$ to both sides of the equation.
4. Factor the left side as the square of binomial.
5. Take the square root on both sides and remember: ${\left(\text{x+q}\right)}^{\text{2}}\text{=r}$ Is equivalent to $\left(\text{x+q}\right)\text{=±}\sqrt{\text{r}}$ .

Calculation:

  ${\text{x}}^{\text{2}}\text{+x+}\frac{\text{1}}{\text{3}}\text{=}\frac{\text{2}}{\text{3}}$ .

  ${\text{x}}^{\text{2}}\text{+2}\left(\text{x}\right)\left(\frac{\text{1}}{\text{x}}\right)\text{+}{\left(\frac{\text{1}}{\text{x}}\right)}^2-{\left(\frac{\text{1}}{\text{x}}\right)}^2+\frac{\text{1}}{\text{3}}-\frac{\text{2}}{\text{3}}=0$ .

  ${\text{x}}^{\text{2}}\text{+x-}\frac{\text{1}}{\text{3}}\text{=0}$ .

Since here the number and variable $\text{x}$ cannot be separated so it can be solved through discriminant method.

Here $\text{a=1,b=1 and c=-}\frac{1}{3}$ .

According to the discriminant method formula:

  $\text{x=}\frac{\text{-b±}\sqrt{{\text{b}}^{\text{2}}\text{-4ac}}}{\text{2a}}$ .

Putting the value of $\text{a,b and c}$ .

  $\text{x=}\frac{\text{-1±}\sqrt{1^{\text{2}}\text{-4}\left(1\right)\left(-\frac{1}{3}\right)}}{\text{2a}}$ .

  $\text{x=}\frac{\text{-1±}\sqrt{1+\frac{4}{3}}}{\text{2}}$ .

  $\text{x=}\frac{\text{-1±}\sqrt{\frac{7}{3}}}{\text{2}}$ .

  $\text{x=}\frac{\text{-1±1}\text{.5275}}{\text{2}}$ .

Hence, $\text{x=-1}\text{.26375}$ and $\text{x=0}\text{.26375}$ .