Chapter 9, Problem 39SGR

To determine

To solve the given equation by completing the square.

Answer to Problem 39SGR

The values of y are -2 and 10 .

Explanation of Solution

Given information :

The function provided is $y^2-8y+16=36$ .

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 :

  $y^2-8y+16=36$

Given equation

  $y^2-8y=20$

Subtracting 16 from both sides .

Here $b=-8$ , the coefficient of y.

  $\frac{b}{2}=-\frac{8}{2}$

  $=-4$

For making the expression a perfect square.

  ${\left(\frac{b}{2}\right)}^2={\left(-4\right)}^2$

  $=16$

This is the value of the constant.

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

  $y^2-8y+16=36$

Adjust the values accordingly

  $=>{\left(y-4\right)}^2=36$

Taking square root.

  $=>{\left(y-4\right)}^2=\sqrt{36}$

Compute the values.

  $=>\left(y-4\right)=\pm 6$

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

  $y-4=-6$

Simplify for y .

  $=>y=-2$

This one value.

  $y-4=6$

Computing the other value.

  $=>y=10$