Question

Square Root

Factor the perfect-square trinomial on the left side of the equation. Apply the square root property of equality. (looking for the answer to the second part)

Answer 1

1. Perfect square trinomial on left sides is
`(x+(1)/(4))^2=(4)/(9)`.

2. The equation after applying the square root property of equality is
`x+(1)/(4)=\pm (2)/(3)`.

Explanation:

The given equation is

`x^2+(1)/(2)x+(1)/(16)=(4)/(9)`

It can be written as

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

Factor the perfect-square trinomial on the left side of the equation.

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

`(x+(1)/(4))^2=(4)/(9)`
`[\because (a+b)^2=a^2+2ab+b^2]`

Therefore the required equation is

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

Taking square root both the sides.

`\sqrt{(x+(1)/(4))^2}=\pm\sqrt{(4)/(9)}`

`x+(1)/(4)=\pm (2)/(3)`

Therefore the equation after applying the square root property of equality is
`x+(1)/(4)=\pm (2)/(3)`.

Answer 2

1).
`(x+(1)/(4))^(2)=(4)/(9)`

2).
`x+(1)/(4)=\pm (2)/(3)`

Explanation:

Here the given equation is
`x^(2)+(1)/(2)x+(1)/(16)=(4)/(9)`

We have to factor the perfect square trinomial on the left side of the equation

So left side of the equation is
`x^(2)+(1)/(2)x+(1)/(16)`

=
`x^(2)+2* (1)/(4)x+((1)/(4))^(2)`

`=(x+(1)/(4))^(2)` since [(a+b)²= a²+b²+2ab]

Therefore the factorial form of the equation will be

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

Now we have to solve the equation by applying square root property

`\sqrt{(x+(1)/(4))^(2)}=\sqrt{(4)/(9)}`

`x+(1)/(4)=\pm (2)/(3)`