Question

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​

Answer 1

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.