I need help please this is due today
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Proof by contradiction.
Let assume that when is an irrational number, then
is a rational number.
If is a rational number, then it can be expressed as a fraction
where
.
Since , the number
is also a rational number. But this contardicts our initial assumption that
is an irrational number. Therefore
must be an irrational number.
To start to solve this problem, we need to know what vertex form is. The vertex form of a parabola is. The vertex form of a parabola is a(x-h) + k, where k is the vertical shift, h is the horizontal shift, and a is the value that tells the stretch.
To start to solve this equation, we want to start to create a difference of two squares.
y = 2(x²+x) We do this step to make the x² have a coefficient of 1
Now, we want to complete the square. To complete the square, we take 1/2 of the coefficient of x, and then square that.
1/2 * 1/2 = 1/4, and 1/4²=1/16
That means that we need to add 1/16 inside and outside the parenthesis.
We get:
y = 2(x²+1/2x + 1/16) - 1/16*2
We do -1/16*2 on the outside because since we added it inside the parenthesis, we need to take it away somewhere else (if that makes sense). The two is there because there is a two in front of the parenthesis.
We get:
y = 2(x+1/4)² - 1/8, by completing the square and simplifying, and this is the final answer.