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Rewrite each relation in the form

y = a * (x - h) ^ 2 + k
by completing the square.
a) y = x ^ 2 + 6x - 1
b) v = x ^ 2 + 2x + 7
c) y = x ^ 2 + 10x + 20
d) v = x ^ 2 + 2x - 1
e) y = x ^ 2 - 6x - 4
f) y = x ^ 2 - 8x - 2
g) y = x ^ 2 - 12x + 8​

User Rahulbmv
by
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1 Answer

3 votes

a.
\[ y = (x^2 + 6x + 9) - 9 - 1 = (x + 3)^2 - 10 \]

b.
\[ v = (x^2 + 2x + 1) + 6 = (x + 1)^2 + 6 \]

c.
\[ y = (x^2 + 10x + 25) - 25 + 20 = (x + 5)^2 - 5 \]

d.
\[ v = (x^2 + 2x + 1) - 1 = (x + 1)^2 - 1 \]

e.
\[ y = (x^2 - 6x + 9) - 9 - 4 = (x - 3)^2 - 13 \]

f.
\[ y = (x^2 - 8x + 16) - 16 - 2 = (x - 4)^2 - 18 \]

g.
\[ y = (x^2 - 12x + 36) - 36 + 8 = (x - 6)^2 - 28 \]

To rewrite each given quadratic relation in the form
\(y = a \cdot (x - h)^2 + k\) by completing the square, we'll follow these steps:

1. Group the
\(x^2\) and \(x\) terms together.

2. Complete the square for the quadratic expression.

3. Rewrite the expression in the desired form.


a) \(y = x^2 + 6x - 1\):\\ y = (x^2 + 6x + 9) - 9 - 1 = (x + 3)^2 - 10 \]

b)
\(v = x^2 + 2x + 7\):


\[ v = (x^2 + 2x + 1) + 6 = (x + 1)^2 + 6 \]

c)
\(y = x^2 + 10x + 20\):


\[ y = (x^2 + 10x + 25) - 25 + 20 = (x + 5)^2 - 5 \]

d)
\(v = x^2 + 2x - 1\):


\[ v = (x^2 + 2x + 1) - 1 = (x + 1)^2 - 1 \]

e)
\(y = x^2 - 6x - 4\):


\[ y = (x^2 - 6x + 9) - 9 - 4 = (x - 3)^2 - 13 \]

f)
\(y = x^2 - 8x - 2\):


\[ y = (x^2 - 8x + 16) - 16 - 2 = (x - 4)^2 - 18 \]

g)
\(y = x^2 - 12x + 8\):


\[ y = (x^2 - 12x + 36) - 36 + 8 = (x - 6)^2 - 28 \]

In each case, the quadratic relation has been successfully rewritten in the desired form by completing the square.

User Muffo
by
7.6k points

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