Question

Rewrite each expression by completing the square.

1. −2x^2 + 8x + 5
2. 2.5????^2 − 7.5???? + 1.25
3. 4 / 3????^2 + 6???? − 5
4. 1000c^2 − 1250c + 695
5. 8n^2 + 2n + 5

Answer 1

1)
`x^2 -4x +((4)/(2))^2 -(5)/(2) - ((4)/(2))^2`

`(x-2)^2 -(5)/(2) -4=(x-2)^2 -(13)/(2)`

2)
`x^2 -3x +((3)/(2))^2 +(1)/(2) - ((3)/(2))^2`

`(x-(3)/(2))^2 +(1)/(2) -(9)/(4)=(x-(3)/(2))^2 -(7)/(4)`

3)
`x^2 +(9)/(2)x +((9)/(4))^2 -(15)/(4) - ((9)/(4))^2`

`(x+(9)/(4))^2 -(15)/(4) -(81)/(16)=(x+(9)/(4))^2 -(141)/(16)`

4)
`c^2 -(5)/(4)c +((5)/(8))^2 -(139)/(200) - ((5)/(8))^2`

`(c-(5)/(8))^2 -(139)/(200) -(25)/(64)=(c-(5)/(8))^2 -(1737)/(1600)`

5)
`n^2 +(1)/(4)n +((1)/(8))^2 +(5)/(8) - ((1)/(8))^2`

`(n+(1)/(8))^2 +(5)/(8) -(1)/(64)=(n+(1)/(8))^2 +(39)/(64)`

Explanation:

1. −2x^2 + 8x + 5

For this case we can begin dividing all the terms by -2 and we got:

`x^2 -4x -(5)/(2)[/tex</p><p>And if we complete the square we got:</p><p>[tex] x^2 -4x +((4)/(2))^2 -(5)/(2) - ((4)/(2))^2`

`(x-2)^2 -(5)/(2) -4=(x-2)^2 -(13)/(2)`

2. 2.5x^2 − 7.5x + 1.25

For this case we can begin dividing all the terms by 2.5 and we got:

`x^2 -3x + (1)/(2)`

And if we complete the square we got:

`x^2 -3x +((3)/(2))^2 +(1)/(2) - ((3)/(2))^2`

`(x-(3)/(2))^2 +(1)/(2) -(9)/(4)=(x-(3)/(2))^2 -(7)/(4)`

3. 4 / 3x ^2 + 6x − 5

For this case we can begin dividing all the terms by 4/3 and we got:

`x^2 + (9)/(2)x - (15)/(4)`

And if we complete the square we got:

`x^2 +(9)/(2)x +((9)/(4))^2 -(15)/(4) - ((9)/(4))^2`

`(x+(9)/(4))^2 -(15)/(4) -(81)/(16)=(x+(9)/(4))^2 -(141)/(16)`

4. 1000c^2 − 1250c + 695

For this case we can begin dividing all the terms by 1000 and we got:

`c^2 - (5)/(4)c + (139)/(200)`

And if we complete the square we got:

`c^2 -(5)/(4)c +((5)/(8))^2 +(139)/(200) - ((5)/(8))^2`

`(c-(5)/(8))^2 -(139)/(200) -(25)/(64)=(c-(5)/(8))^2 -(487)/(1600)`

5. 8n^2 + 2n + 5

For this case we can begin dividing all the terms by 8 and we got:

`n^2 + (1)/(4)x + (5)/(8)`

And if we complete the square we got:

`n^2 +(1)/(4)n +((1)/(8))^2 +(5)/(8) - ((1)/(8))^2`

`(n+(1)/(8))^2 +(5)/(8) -(1)/(64)=(n+(1)/(8))^2 +(39)/(64)`