Chapter 9.4, Problem 61E

To determine

To solve the equation by completing the square method.

Answer to Problem 61E

Value of x are $x=2-\sqrt{5.25}$ and $x=2+\sqrt{5.25}$

Explanation of Solution

Given:

Equation:

  $\frac{8}{3}x-\frac{2}{3}{x}^2=-\frac{5}{6}$

Calculation:

According to question equation is,

  $-\frac{2}{3}{x}^2+\frac{8}{3}x=-\frac{5}{6}$

Changing the sign,

  $\frac{2}{3}{x}^2-\frac{8}{3}x=\frac{5}{6}$

To make coefficient of $x^2$ 1 we need to divide the whole equation $\frac{2}{3}$

So,

  $\frac{\frac{2}{3}{x}^2}{\frac{2}{3}}-\frac{\frac{8}{3}x}{\frac{2}{3}}=\frac{\frac{5}{6}}{\frac{2}{3}}$

On solving,

  $x^2-4x=\frac{5}{4}$

Taking square half of coefficient of $x$

i.e. ${(4\times \frac{1}{2})}^2$

so,

  ${(2)}^2=4$

Adding this on both side,

  $x^2-4x+4=\frac{5}{4}+4$

  $x^2-4x+4$ can be written as ${(x-2)}^2$

Therefore,

  ${(x-2)}^2=\frac{5}{4}+4$

Solving right hand side,

  ${(x-2)}^2=\frac{5+16}{4}$

So,

  $x-2=\sqrt{\frac{21}{4}}$

Therefore ,

  $x-2=\sqrt{5.25}$

Finding the value of $x$

  $x-2=\pm \sqrt{5.25}$

So,

  $x=2\pm \sqrt{5.25}$

Value of x are $x=2-\sqrt{5.25}$ and $x=2+\sqrt{5.25}$