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# What is the sum of all unique solutions for x^2+6x+9=|x+3| ?

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What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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21 Apr 2017, 07:46
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Question Stats:

56% (02:02) correct 44% (02:01) wrong based on 502 sessions

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What is the sum of all unique solutions for $$x^2+6x+9=|x+3|$$ ?

A. -12
B. -9
C. -7
D. -5
E. 0

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Re: What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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21 Apr 2017, 08:20
5
1
What is the sum of all unique solutions for x2+6x+9=|x+3|x2+6x+9=|x+3| ?

A. -12
B. -9
C. -7
D. -5
E. 0

eq. 1 : x2 + 6x + 9 = x+3
x2 + 5x + 6 = 0

solves x = -3, -2.

eq. 2 : x2 + 6x + 9 = - (x+3)

x2 + 7x + 12 = 0

solves x = -4,-3

Unique soln. -3, -4 ,-2

Sum = -9.

Ans. B
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Re: What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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21 Apr 2017, 08:19
LHS = ( x+3 )^2
RHS is absolute value,,

square both RHS and LHS,, n den factorise.. im stuck then,,,any help plz
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Re: What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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21 Apr 2017, 08:34
mohshu wrote:
LHS = ( x+3 )^2
RHS is absolute value,,

square both RHS and LHS,, n den factorise.. im stuck then,,,any help plz

(x+3)^2 = |x+3|
if x>=-3
(x+3)^2 = x+3 => x = -3 or -2
if x<-3
(x+3)^2 = -x-3 => x=-4
sum of all the roots of x =-3-2-4=-9
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Re: What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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22 Apr 2017, 03:09
What is the sum of all unique solutions for x^2+6x+9=|x+3| ?

A. -12
B. -9
C. -7
D. -5
E. 0

x^2+6x+9= x+3 (Note - considering X+3 non negative only possible if X > -3)
x^2+5x+6= 0 X= -2

x^2+6x+9= -(x+3)
x^2+7x+12= 0 X= -4 or -3

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Re: What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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22 Apr 2017, 03:48
IMO B
As LHS has inequality we have to take both positive and negative signs for LHS.
Solving for that we will get sum of -9
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Re: What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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22 Apr 2017, 11:55
Bunuel wrote:
What is the sum of all unique solutions for $$x^2+6x+9=|x+3|$$ ?

A. -12
B. -9
C. -7
D. -5
E. 0

Here is my approach
Eq. 1
x^2+6x+9 = x + 3
x^2+5x+6 = 0
(x+3)(x+2)=0
x1 = -3 x2=-2

Eq. 2
x^2+6x+9 = -x+3
x^2+7x+6 = 0
(x+1)(x+6) = 0
x1=-1 x2=-6

The sum = -3-2-1-6=-12 answer: (A)

Please correct me if I'm wrong

regards.
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What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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22 Apr 2017, 23:40
maxbm10 wrote:
Bunuel wrote:
What is the sum of all unique solutions for $$x^2+6x+9=|x+3|$$ ?

A. -12
B. -9
C. -7
D. -5
E. 0

Here is my approach
Eq. 1
x^2+6x+9 = x + 3
x^2+5x+6 = 0
(x+3)(x+2)=0
x1 = -3 x2=-2

Eq. 2
x^2+6x+9 = -x+3
x^2+7x+6 = 0
(x+1)(x+6) = 0
x1=-1 x2=-6

The sum = -3-2-1-6=-12 answer: (A)

Please correct me if I'm wrong

regards.

I think you forgot the distribute the negative sign for
X^2 +6x+9= -(x+3)

It should result in

X^2+7x+12
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Re: What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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23 Apr 2017, 00:19
x^2+6x+9=|x+3|
(x+3)^2=|x + 3|
but (x+3)^2 ≥0
therefore, x^2+6x+9=(x+3) and x^2+6x+9 cannot be equal to -(x+3)
solving, x^2+6x+9=(x+3) will give us x=-2 and x=-3
therefore ans is x=-5
option d
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Re: What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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27 Apr 2017, 19:16
Can someone please explain how the equation got to x^2+5x+6?
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What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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27 Apr 2017, 19:44
1
|x| can take value either +x or -x. We need to consider both the options if nothing is specified about x

So considering two possibilties of the equation
$$x^2$$+6x+9 = x+3 &&& $$x^2$$+6x+9=-(x+3)
$$x^2$$+6x+9 - x - 3 = 0 &&& $$x^2$$+6x+9 + x+3 = 0
$$x^2$$+5x+6=0 &&& $$x^2$$+7x+12=0
(x+3)(x+2)=0 &&& (x+3)(x+4) = 0
x=-3,-2 &&& x=-3,-4

Total Sum = -12

But we need unique solutions' sum so -3+-2+-4=-9
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Re: What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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10 May 2017, 01:17
Bunuel wrote:
What is the sum of all unique solutions for $$x^2+6x+9=|x+3|$$ ?

A. -12
B. -9
C. -7
D. -5
E. 0

First, the left-hand side of the equation should be factored out:

(x+3)(x+3)=lx +3l

take the opposite value of the value inside each parentheses and add

(-3) + (-3)

(0)(0)= lx+3l

only -3 can satisfy the equation so add -3 to -3 + -3

sum = -9

Hence "B"
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What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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10 May 2017, 03:55
1
Bunuel wrote:
What is the sum of all unique solutions for $$x^2+6x+9=|x+3|$$ ?

A. -12
B. -9
C. -7
D. -5
E. 0

We have $$x^2+6x+9=(x+3)^2=|x+3|$$

Solution 1.

If $$x \geq -3 \implies |x+3|=x+3$$
$$\implies (x+3)^2=x+3 \implies (x+3)^2-(x+3)=0 \implies (x+3)(x+2)=0 \implies x =-3 \, \, \, \text{or} \, \, \, x=-2$$

If $$x \leq -3 \implies |x+3|=-(x+3)$$
$$\implies (x+3)^2=-(x+3) \implies (x+3)^2+(x+3)=0 \implies (x+3)(x+4)=0 \implies x =-4$$

All roots for this equation are $$\{ -2, -3, -4 \}$$. The sum of these roots are $$-9$$. The answer is B.

Solution 2.

$$(x+3)^2=|x+3| \iff |x+3|^2-|x+3|=0 \iff |x+3|(|x+3|-1)=0 \implies |x+3| =0 \, \, \text{or} \, \, |x+3|=1$$
$$|x+3|=0 \implies x=-3$$
$$|x+3|=1 \implies x=-2 \, \, \text{or} \, \, x=-4$$.

The same result as Solution 1.
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Re: What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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12 May 2017, 13:37
Bunuel wrote:
What is the sum of all unique solutions for $$x^2+6x+9=|x+3|$$ ?

A. -12
B. -9
C. -7
D. -5
E. 0

We can solve the given absolute equation for when |x + 3| is positive and for when |x + 3| is negative.

|x + 3| is positive:

x^2 + 6x + 9 = x + 3

x^2 + 5x + 6 = 0

(x + 2)(x + 3) = 0

x = -2 or x = -3

|x + 3| is negative:

x^2 + 6x + 9 = -x - 3

x^2 + 7x + 12 = 0

(x + 3)(x + 4) = 0

x = -3 or x = -4

Thus, the sum of the unique values of x is -2 - 3 - 4 = -9

Alternate Solution:

We recognize that the right hand side is the expansion of (x + 3)^2; thus:

(x + 3)^2 = |x + 3|

Let’s substitute x + 3 = u. Then, the equation becomes

u^2 = |u|

We are looking for numbers whose square is equal to its absolute value. This is only possible if u = 0, 1 or -1. Since u = x + 3, this means x + 3 = 0, x + 3 = 1 or x + 3 = -1. These equations give us the solutions x = -3, x = -2 and x = -4, respectively. Thus, the sum of the unique solutions is -3 + (-2) + (-4) = -9

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What is the sum of all unique solutions for x^2+6x+9=|x+3| ?  [#permalink]

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03 Aug 2017, 08:34
mrdlee23 wrote:
Can someone please explain how the equation got to x^2+5x+6?

When you have absolute values there are 2 equations since the absolute value can equal a negative or positive number. Essentially, the absolute value is a distance from the origin.

So (X^2)+6X+9 = positive X+3
AND (X^2) +6X+9 = Negative X+3

So X^2+6X+9 = positive X+3 so that is (X^2) +6X+9=x+3 or (X^2)+5X+6=0 Solves into (x+3)(x+2)

And AND X^2 +6X+9 = Negative X+3 = (X^2) +6X +9 = -(X+3) or (X^2)+6X+9=-x-3 which equals (X^2) +7X+12. Solves into (X+3) and (x+4)

So right now we are feeling pretty good we have solutions -4,-3 and -2 however there could be one last trick so we have to ensure that the left side of the equation = a non negative value so check -4,-3,-2 in the equation (X^2)+6X+9 Notice the 3 solutions add to 1,0 & 1 so all equal a non negative value.

The ABS value right side can be either negative or positive but the left side (in this case but I mean the non absolute value side) of the equation must be a non negative number. Why is that? Because the absolute value is a function which always shoots out a non negative value so |-100| = 100 |-25| = 25 .|100| = 100, |0| =0. So if the LHS of the equation equaled a negative number then it couldn't equal the RHS which is positive. That is the trick they didn't test here but it's important to note.

One more comparison

|-100|=100
but
|-100| does not = -100 as |-100|=100
What is the sum of all unique solutions for x^2+6x+9=|x+3| ?   [#permalink] 03 Aug 2017, 08:34
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