Chapter 9, Problem 40SGR

To determine

To solve the given equation by completing the square.

Answer to Problem 40SGR

The values of y are 0.4 and 5.6 .

Explanation of Solution

Given information :

The function provided is $y^2-6y+2=0$ .

Formula used :

To solve an equation by completing the square, divide the coefficient of y of the equation by 2 and square it. That should provide the constant value. Then the equation would be of the form ${\left(x+\frac{b}{2}\right)}^2$ . Adjust the other constants and then square root the left-hand and right-hand sides. This would give the values of the equation.

Calculation :

  $y^2-6y+2=0$

Given equation

Here $b=-6$ , the coefficient of y.

  $\frac{b}{2}=-\frac{6}{2}$

  $=-3$

For making the expression a perfect square.

  ${\left(\frac{b}{2}\right)}^2={\left(-3\right)}^2$

  $=9$

This is the value of the constant.

Make left hand side a perfect square using the value of b .

  $y^2-6y+2+7=7$

  $=>y^2-6y+9=7$

Adjust the values accordingly

  $=>{\left(y-3\right)}^2=7$

Taking square root.

  $=>\left(y-3\right)=\sqrt{7}$

Compute the values.

  $=>\left(y-3\right)=\pm 2.6$

Now, equate both values, to get the answers of y.

  $y-3=-2.6$

Simplify for y .

  $=>y=0.4$

This one value.

  $y-3=2.6$

Computing the other value.

  $=>y=5.6$