Chapter 9, Problem 44SGR

To determine

To solve the given equation by completing the square.

Answer to Problem 44SGR

The values of y are 0.29 and 1.71 .

Explanation of Solution

Given information :

The function provided is $2a^2-4a+1=0$ .

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 :

  $2a^2-4a+1=0$

Given equation

  $\frac{2a^2-4a+1}{2}=\frac{0}{2}$

Making the coefficient of $a^2$ equal to 1.

  $a^2-2a+\frac{1}{2}=0$

  $=>a^2-2a=-\frac{1}{2}$

Making the equation in the standard form .

Here $b=-2$ , the coefficient of a.

  $\frac{b}{2}=-\frac{2}{2}$

  $=-1$

For making the expression a perfect square.

  ${\left(\frac{b}{2}\right)}^2={\left(-1\right)}^2$

  $=1$

This is the value of the constant.

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

  $a^2-2a+1=-\frac{1}{2}+1$

Adjust the values accordingly

  $=>{\left(a-1\right)}^2=0.5$

Taking square root.

  $=>\left(a-1\right)=\sqrt{0.5}$

Compute the values.

  $=>\left(a-1\right)=\pm 0.71$

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

  $a-1=-0.71$

Simplify for a .

  $=>a=0.29$

This one value.

  $=>a-1=0.71$

Computing the other value.

  $=>a=1.71$