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:

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

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

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

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