Question

solve each equation by completing the square.
$0.5x^2 - 11x + 60\frac{1}{2} = 0$
after completing the square, the equation is \boxed{} and the solution is
example: $x=3; x=5$ .

Explanation:

Step1: Multiply by 2 to eliminate decimals and fractions

Multiply the entire equation by 2 to get rid of the decimal and the fraction.
$$2\times(0.5x^{2}-11x + 60\frac{1}{2})=2\times0$$
$$x^{2}-22x + 121 = 0$$

Step2: Rewrite in perfect square form

Notice that \(x^{2}-22x + 121\) is a perfect square trinomial. Recall that \((a - b)^2=a^{2}-2ab + b^{2}\). Here, \(a = x\), \(2ab=22x\) so \(b = 11\), and \(b^{2}=121\). So we can write:
$$(x - 11)^{2}=0$$

Step3: Solve for x

Take the square root of both sides:
\(x-11 = 0\)
Then, add 11 to both sides:
\(x=11\) (since both solutions are the same, we have a repeated root)

Answer:

After completing the square, the equation is \((x - 11)^{2}=0\) and the solution is \(x = 11\) (or \(x=11;x = 11\) to show the repeated root format as in the example)