Chapter 9, Problem 41SGR

To determine

To solve the given equation by completing the square.

Answer to Problem 41SGR

The values of n are - 0.7 and 7.7 .

Explanation of Solution

Given information :

The function provided is $n^2-7n=5$ .

Formula used :

To solve an equation by completing the square, divide the coefficient of n 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 :

  $n^2-7n=5$

Given equation

Here $b=-7$ , the coefficient of n.

  $\frac{b}{2}=-\frac{7}{2}$

For making the expression a perfect square.

  ${\left(\frac{b}{2}\right)}^2={\left(-\frac{7}{2}\right)}^2$

  $=\frac{49}{4}$

This is the value of the constant.

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

  $n^2-7n+\frac{49}{4}=5+\frac{49}{4}$

Adjust the values accordingly

  $=>{\left(n-\frac{7}{2}\right)}^2=\frac{69}{4}$

Taking square root.

  $=>\left(n-\frac{7}{2}\right)=\sqrt{\frac{69}{4}}$

Compute the values.

  $=>n-3.5=\pm 4.2$

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

  $=>n-3.5=4.2$

Simplify for y .

  $=>n=7.7$

This one value.

  $=>n-3.5=-4.2$

Computing the other value.

  $=>n=-0.7$