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# If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x =

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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
Bunuel wrote:
If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x =

(A) -3
(B) -1/2
(C) 0
(D) 1/2
(E) 3/2

x(2x + 1) = 0 --> x=0 OR x=-1/2;
(x + 1/2)(2x - 3) = 0 --> x=-1/2 OR x=3/2.

x=-1/2 satisfies both equations.

Can you show how you manipulated both of the equations to get to zero, and to -1/2, and 3/2, thanks.
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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
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selfishmofo wrote:
Bunuel wrote:
If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x =

(A) -3
(B) -1/2
(C) 0
(D) 1/2
(E) 3/2

x(2x + 1) = 0 --> x=0 OR x=-1/2;
(x + 1/2)(2x - 3) = 0 --> x=-1/2 OR x=3/2.

x=-1/2 satisfies both equations.

Can you show how you manipulated both of the equations to get to zero, and to -1/2, and 3/2, thanks.

When a product of two multiples is 0, it means that either of the multiples (or both) is 0.

$$x(2x + 1) = 0$$ --> $$x=0$$ or $$2x+1=0$$ ($$x=-\frac{1}{2}$$).

$$(x + \frac{1}{2})(2x - 3) = 0$$ --> $$x+\frac{1}{2}=0$$ ($$x=-\frac{1}{2}$$) or $$2x - 3=0$$ ($$x=\frac{3}{2}$$).

Hope it's clear.
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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
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Bunuel wrote:
If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x =

(A) -3
(B) -1/2
(C) 0
(D) 1/2
(E) 3/2

x(2x + 1) = 0 --> x=0 OR x=-1/2;
(x + 1/2)(2x - 3) = 0 --> x=-1/2 OR x=3/2.

x=-1/2 satisfies both equations.

This makes complete sense, although, I ran into trouble when I tried to FOIL the second equation and ended up with x^2-x-3/4=0 and from that point forward, I was completely stumped. Why is that method wrong?

I notice that I get confused on that front quite a bit - FOIL'ing vs. just setting both parenthesis to 0?

EDIT: As I was doing other problems, I ran into DS 67, Pg 180 of OG 13. The equation there is n(n+1) = 6, if I use the same methodology outlined above, the two solutions I get are n=6 and n=5. That is obviously wrong and I should've opted to FOIL in the above case. Hence my confusion -- why is it that in some situations I need to FOIL and in some other situations, I need to just equate the left to the right side WITHOUT foiling?
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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
$$(x + \frac{1}{2})(2x - 3) = 0$$

Multiply both sides by 2

(2x+1)(2x-3) = 0............ (1)

The other equation

x (2x+1) = 0 ............. (2)

With RHS 0 of both the equations, LHS has 2x+1 in common

Equating to 0

x $$= -\frac{1}{2}$$

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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
russ9 wrote:
Bunuel wrote:

x(2x + 1) = 0 --> x=0 OR x=-1/2;
(x + 1/2)(2x - 3) = 0 --> x=-1/2 OR x=3/2.

x=-1/2 satisfies both equations.

This makes complete sense, although, I ran into trouble when I tried to FOIL the second equation and ended up with x^2-x-3/4=0 and from that point forward, I was completely stumped. Why is that method wrong?

I notice that I get confused on that front quite a bit - FOIL'ing vs. just setting both parenthesis to 0?

EDIT: As I was doing other problems, I ran into DS 67, Pg 180 of OG 13. The equation there is n(n+1) = 6, if I use the same methodology outlined above, the two solutions I get are n=6 and n=5. That is obviously wrong and I should've opted to FOIL in the above case. Hence my confusion -- why is it that in some situations I need to FOIL and in some other situations, I need to just equate the left to the right side WITHOUT foiling?

Hi Bunuel,

Still a little confused about the above question. I ran into countless more errors over the past few days -- cause was the same reason mentioned above. Would greatly appreciate some clarification.

Thanks!
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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
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Multiplication of any number with 0 gives 0 ..

When product of two or more expressions is zero, then any expression or product of combination of expressions can be 0.

Here,
x(2x+1) = 0
here product of x and 2x+1 equals 0 => either x = 0 or 2x+1 = 0 or both

hence x=0 or -1/2 will satisfy the first equation.

(x+1/2)(2x-3) = 0
here product of x+1/2 and 2x-3 equals 0=> either x+1/2 = 0 or 2x-3 = 0 or both

hence x = -1/2 or 3/2 will satisfy the second equation.
-1/2 is common in both sets hence x= -1/2 will satisfy both equations.

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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
Bunuel wrote:
If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x =

(A) -3
(B) -1/2
(C) 0
(D) 1/2
(E) 3/2

$$x(2x + 1) = 0$$ --> $$x=0$$ OR $$x=-\frac{1}{2}$$;

$$(x + \frac{1}{2})(2x - 3) = 0$$ --> $$x=-\frac{1}{2}$$ OR $$x=\frac{3}{2}$$.

$$x=-\frac{1}{2}$$ satisfies both equations.

Should we apply the same logic to data sufficiency questions if we are asked to find the value of x and given each quadratic as one of the statements? Thus, statement 1 and statement 2 would each be insufficient but combining together they are sufficient (i.e. the answer is C) due to one shared solution for both quadratics/statements?
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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
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tigrr49 wrote:
Bunuel wrote:
If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x =

(A) -3
(B) -1/2
(C) 0
(D) 1/2
(E) 3/2

$$x(2x + 1) = 0$$ --> $$x=0$$ OR $$x=-\frac{1}{2}$$;

$$(x + \frac{1}{2})(2x - 3) = 0$$ --> $$x=-\frac{1}{2}$$ OR $$x=\frac{3}{2}$$.

$$x=-\frac{1}{2}$$ satisfies both equations.

Should we apply the same logic to data sufficiency questions if we are asked to find the value of x and given each quadratic as one of the statements? Thus, statement 1 and statement 2 would each be insufficient but combining together they are sufficient (i.e. the answer is C) due to one shared solution for both quadratics/statements?

Technically, yes, C would be your answer in a similarly worded DS problem. But beware that not all quadratic equations give you 2 distinct values.

Lets say, statement 1 gave you:

$$x^2+2x+1$$ = 0 , this is in fact $$(x+1)^2$$ ---> $$(x+1)^2 = 0$$---> $$x = -1$$ .

So you get ONLY 1 value and thus this statement will be sufficient on its own. Thus, in DS questions, you should be making sure that you actually are getting 2 distinct values.
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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
I do understant the Method given by Bunuel, but I tried to look for different solution and came to the following :

Equation 1 : x(2x+1)=0 => 2x^2 +x=0 => 2x^2=-x

Equation 2: ( X+1/2)(2x+3) after FOIL => 2x^2 -2x-3/2 , and here I replaced 2x^2 with -x from the first equation and got
-x-2x-3/2=o => -3x=3/2 => x=-1/2

I would like kindly ask Bunuel or other experts if this is also a correct solution?
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If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
kzivrev wrote:
I do understant the Method given by Bunuel, but I tried to look for different solution and came to the following :

Equation 1 : x(2x+1)=0 => 2x^2 +x=0 => 2x^2=-x

Equation 2: ( X+1/2)(2x+3) after FOIL => 2x^2 -2x-3/2 , and here I replaced 2x^2 with -x from the first equation and got
-x-2x-3/2=o => -3x=3/2 => x=-1/2

I would like kindly ask Bunuel or other experts if this is also a correct solution?

Yes, this is a correct method but a bit more time consuming method. In GMAT, you must do proper time management and not just solve the questions correctly.

When you are given a*b =0 ---> 3 cases possible

1. a=0 and b $$neq$$ 0

2. b=0 and a $$neq$$ 0

3. a=b=0

When you are directly given x(2x + 1) = 0 --> either x=0 or 2x+1 =0 --> x=-0.5.

Similarly from the second equation, (x + 1/2)(2x - 3) = 0 ---> either x+0.5 =0 --> x=0 or 2x-3 =0 -->x=1.5.

From these 2 sets of solutions, you see that x=-0.5 is the common solution and is hence the value of x asked in the question.

Hope this helps.
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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
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If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x =

(A) -3
(B) -1/2
(C) 0
(D) 1/2
(E) 3/2

To solve we will use the zero product property. The zero product property states that if the product of two quantities is equal to 0, then at least one of the quantities has to be equal to 0. That is, if a  b = 0, then either a = 0 or b = 0. Of course, both a and b can be 0 at the same time. The point is that at least one of them has to be 0.

Let’s start determining the value(s) of x in the equation x(2x + 1) = 0

If x(2x + 1) = 0, we know:

x = 0

OR

2x + 1 = 0

2x = -1

x = -1/2

Thus, x = 0 or x = -1/2

Let’s now determine the value(s) of x in the second equation (x + 1/2)(2x - 3) = 0

(x + 1/2)(2x - 3) = 0, we know:

(x + 1/2) = 0

x = -1/2

OR

(2x - 3) = 0

2x = 3

x = 3/2

Thus, x = -1/2 or x = 3/2

Because we need to determine a value for x in both equations, the answer is x = -1/2.

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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
x = 0 or x = -1/2
and
x = -1/2 or x = 3/2

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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
Hi all ,

I ended up by solving the two quadratic equations simultaneously to find the value of x instead of solving one by one for the value of x .

Posted from my mobile device
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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
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LeenaSai wrote:
Hi all ,

I ended up by solving the two quadratic equations simultaneously to find the value of x instead of solving one by one for the value of x .

Posted from my mobile device

LeenaSai

You don’t need to solve for quadratic here. It’s already given in solved form.

If a quadratic equation after factorisation would look like (x - a)(x - b) = 0
—> x - a = 0 or x - b = 0
—> x = a or b

x(2x + 1) = 0
—> x = 0 or 2x + 1 = 0
—> x = 0 or -1/2

(x + 1/2)(2x - 3) = 0
—> x + 1/2 = 0 or 2x - 3 = 0
—> x = -1/2 or x = 3/2

Combining 1st and 2nd equation, we get common value of x as -1/2

So, Option B
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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x =

(A) -3
(B) -1/2
(C) 0
(D) 1/2
(E) 3/2

x(2x+1)=0
x=0 or -1/2

(x+1/2)(2x-3)=0
x=-1/2 or 3/2

x=-1/2 is common to both equations

IMO B

Posted from my mobile device
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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
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Video solution from Quant Reasoning:
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Re: If x(2x + 1) = 0 and (x + 1/2)(2x - 3) = 0, then x = [#permalink]
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Hi All,

This question gives us a couple of Quadratic equations that have already been factored:

(X)(2X + 1) = 0 and (X + ½)(2X – 3) = 0

We’re asked when X equals. This is a standard mid-level Algebra prompt that has already done half the work for us, so we just need to calculate the two values of X in each equation and find the one answer that shows up in BOTH equations.

With (X)(2X + 1) = 0

X = 0 and X = -1/2 are the two solutions

With (X + ½)(2X – 3) = 0

X = -1/2 and X = 3/2 are the two solutions

Based on these results, the answer that shows up in BOTH is…