Question

Complete the square rational only:Solve the equation for all values of x by completing the square.x^2 + 4x + 3 = 0

Answer 1

Problem:

Solve the equation for all values of x by completing the square.



x^2 + 4x + 3 = 0

remember the following:

To solve

`ax^2\text{ + bx +c= }0`

by completing the square, we carry out the following steps:

1. Transform the equation so that the constant term, c, is only on the right-hand side. In our case:

`x^2\text{ + 4x = -3}`

2. If a , the leading coefficient (the coefficient of the term x^2), is not equal to 1, divide both sides by a. Notice that in our case a = 1. Then, we don't do this step.

3. Add (b/2a)^2 to the right and the left side of the equation, that is:

`x^2\text{ + 4x + (}(b)/(2a))^2\text{= -3 + (}(b)/(2a))^2`

in our case note that b = 4 and a = 1, so we have:

`x^2\text{ + 4x + (}(4)/(2))^2\text{= -3 + (}(4)/(2))^2`

this is equivalent to say:

`x^2\text{ + 4x + (2})^2\text{= -3 + (2})^2\text{ = -3+4}`

this is equivalent to say

`x^2\text{ + 4x + (2})^2=1`

this is equivalent to say:

`(x+2)^2=1`

now, we take the square root of both sides of the equation

`\sqrt[]{(x+2)^2}=\sqrt[]{1}`

this is equivalent to say:

`x+2^{}=\pm1`

solve for x

`x=\pm1\text{ - 2}`

then, we have two solutions:

`x=1-2\text{ = -1}`

and

`x=-1-2\text{ = -}3`

the two roots(zeros) are x = -1 and x = -3