Respuesta :

Completing Squares

It's given the following equation:

[tex]x^2-20x=-2x-80[/tex]

We are required to express the equation in the form:

[tex](x+a)^2=b[/tex]

The first step is sending all the variables to the left side of the equation.

Adding 2x:

[tex]\begin{gathered} x^2-20x+2x=-80 \\ \\ \text{Simplifying:} \\ x^2-18x=-80 \end{gathered}[/tex]

To complete squares, we need to recall the following identity:

[tex]p^2+2pq+q^2=(p+q)^2[/tex]

The expression on the left side is missing the third term to be a perfect square. Note that comparing

p=x

2pq = -18x

This means that

q = -18x/2p

q = -18x/2x

q = -9

Now we know the value of the second term, we need to add q^2=81:

[tex]x^2-18x+81=-80+81[/tex]

The left side of the equation is the square of x-9, and the right side can be calculated:

[tex](x-9)^2=1[/tex]

Now we have the required expression, where a=-9 and b = 1

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