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Which of the following equations has a root in common with x^2 - 6x +

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Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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Which of the following equations has a root in common with \(x^2 - 6x + 5 = 0\)?

A. \(x^2 + 1 = 0\)

B. \(x^2 - x - 2 = 0\)

C. \(2x^2 - 2 = 0\)

D. \(x^2 - 2x - 3 = 0\)

E. \(x^2 - 10x - 5 = 0\)
[Reveal] Spoiler: OA
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Re: Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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New post 06 Feb 2011, 00:38
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x^2-6x+5=0
(x-1)(x-5)=0
roots are 1 and 5.

Substitute these roots in all the equations;

1.x^2+1=0
x=1; 1^2+1=1+1=2!=0. Not a root.
x=5; 5^2+1=25+1=26!=0. Not a root.

2.x^2-x-2=0
x=1; 1^2-1-2=-2!=0. Not a root.
x=5; 5^2-5-2=25-7=18!=0. Not a root.

3.x^2-10x-5=0
x=1; 1^2-10*1-5=-14!=0. Not a root.
x=5; 5^2-10*5-5=25-55=-30!=0. Not a root.

4.2x^2-2=0
x=1; 2*1^2-2=2-2=0=0. 1 is a root. We can stop here.
x=5; 2*5^2-2=50-2=48!=0. Not a root.

5. x^2-2x-3=0
x=1; 1^2-2*1-3=-3!=0. Not a root.
x=5; 5^2-2*5-3=12!=0. Not a root.

Ans: "D"

Let's expand the correct answer algebraically;
2x^2-2=0
2*x^2=2
x^2=1
x=+1 and x=-1

even your answer seems correct;
(2x-2)(x+1) = 0
So; 2x-2=0; x=2/2=1. "1 is also one of the roots in the equation(x^2-6x+5=0)"
x+1=0; x=-1.
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Re: Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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New post 06 Feb 2011, 04:21
I like more -finding roots and further substitution as you explained first. I cant understand second solving algebraically, because other equations also share x=1 root.

I appreciate your help. Thanks a lot
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Re: Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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New post 06 Feb 2011, 05:18
Lolaergasheva wrote:
I like more -finding roots and further substitution as you explained first. I cant understand second solving algebraically, because other equations also share x=1 root.

I appreciate your help. Thanks a lot


None of the other equations has either 1 or 5 as root.

The quadratic equations in your example were not factored properly.

1.
x^2+1=0
was factored as
"(x-1)(x+1)", which is not correct

(x-1)(x+1) = x^2 - 1^2 = x^2 - 1 and not {x^2 + 1}

The correct way of factoring it is;
x^2 = -1
Well I see that; it cannot be factored because \(x = \sqrt{-1}\) will result in some imaginary number.
So; x^2+1=0 has zero roots.

2.
x^2-x-2=0
x^2-2x+x-2=0
x(x-2)+1(x-2)=0
(x+1)(x-2)=0
So roots are;
x+1=0; x=-1 (Not 1)
x-2=0; x=2

3.
x^2-10x-5=0
This has two solutions but cannot be factored like other equations;
We will have to use discriminant approach to find roots;

For a quadratic equation; ax^2+bx+c=0
The two roots are;
\(Roots=\frac{-b \pm sqrt{b^2-4*a*c}}{2*a}\)

x^2-10x-5=0
can be written as
1x^2+(-10)x+(-5)=0
a=1
b=-10
c=-5

Plug these values in the formula;
\(Roots = \frac{-(-10) \pm sqrt{(-10)^2-4*1*(-5)}}{2*1}\)
\(Roots = \frac{10 \pm sqrt{100+20}}{2*1}\)
\(Roots = \frac{10 \pm sqrt{120}}{2}\)
\(Roots=5 \pm \frac{sqrt{120}}{2}\)

###\(\frac{sqrt{120}}{2} \approx 5.5\)###

\(Root_1 \approx 5+5.5=10.5\)
\(Root_2 \approx 5-5.5=-0.5\)

Neither of the roots is 1 or 5.

4.
2x^2-2=0
2(x^2-1)=0
x^2-1=0
x^2=1
i.e. x=1(This is the only place where the root is 1) and x=-1

5.
x^2-2x-3=0
x^2-3x+x-3=0
x(x-3)+1(x-3)=0
(x+1)(x-3)=0
x+1=0; x=-1(Not 1)
x-3=0; x=3

Please visit the following link should you need in-depth theory about quadratic equation and factorization;

http://gmatclub.com/forum/algebra-101576.html#p787276
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Re: Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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New post 06 Feb 2011, 06:58
Thank you for the link and for correcting my mistakes.
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Re: Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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New post 06 Feb 2011, 10:02
Roots of x^2-6x+5=0 are 1 and 5
After scanning the answer choices, I chose easy equation 2x^2-2=0 which gives x = 1 or -1 So D
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Re: Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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New post 14 Nov 2011, 07:53
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Easy question. \(x^2 - 6x + 5 = 0\) factors into \((x-1)(x-5)=0\). Thus, roots are x=1 and x=5.

Do the same for each option:

A. \(x^2 + 1 = 0\)
\(x^2 = -1\)
roots are imaginary

B. \(x^2 - x - 2 = 0\)
\((x-2)(x+1) = 0\)
roots are x=2 and x=-1
none match question


C. \(2x^2 - 2 = 0\)
\(2(x^2-1)=0\)
\((x^2-1) = 0\)
\((x-1)(x+1) = 0\)
roots are x=1 and x=-1
x=1 matches the root of original equation

No need to do the rest, we have our answer. However, if you have lots of time you may want to do them anyway to make sure you didn't make a mistake above. You'll know you made a mistake somewhere if any of the questions below also yield a root in common.

D. \(x^2 - 2x - 3 = 0\)
\((x-3)(x+1) = 0\)
roots are x=3 and x=-1, no roots in common

E. \(x^2 - 10x - 5 = 0\)
doesn't factor into integers, no roots in common

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Re: Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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New post 14 Nov 2011, 14:20
ok here is how I approached the problem:

I solved the equation to get the x value
x=5, x=1

and I solved each of the equations to see if any of the equations has a common factor with the equation in the stem quation.
non of the had anything in common except for C, which has x=1,x=-1.

hope that helps
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Re: Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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New post 14 Nov 2011, 21:56
Stoneface wrote:
Which of the following equations has a root in common with x^2 - 6x + 5 = 0?

A. x^2 + 1 = 0
B. x^2 - x - 2 = 0
C. 2x^2 - 2 = 0
D. x^2 - 2x - 3 = 0
E. x^2 - 10x - 5 = 0


The given equation can be simplified to (x-1)(x-5)=0
Roots are 1 and 5.

You don't need to find roots of each equation give as options
Just simplify each option

After quick inspection only E can have 5 as a probable root. So only check whether or not 1 is a root.

A. No real root
B. (x-2)(x+1)=0 [x=1 does not make the left hand side zero]
C. 2(x^2-1)=0 [x=1 does make the left hand side zero]
D. (x-3)(x+1)=0 [x=1 does not make the left hand side zero]
E. Don't care... by observation neither 1 nor 5 can be a root

Hence C
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Re: Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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New post 28 Dec 2015, 12:58
Which of the following has "A" root common. Only C has one root common with the equation in the question.
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Re: Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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New post 08 Jul 2016, 01:00
Lolaergasheva wrote:
Which of the following equations has a root in common with x^2 - 6x + 5 = 0?

A. x^2 + 1 = 0
B. x^2 - x - 2 = 0
C. 2x^2 - 2 = 0
D. x^2 - 2x - 3 = 0
E. x^2 - 10x - 5 = 0


x^2 - 6x + 5 = 0
(x-5)(x-1) = 0
The roots are 1 and 5

We need to check for each option to calculate the roots. This can be done by plugging in the values in the equations

(A) x2 + 1 = 0. None of 1 or 5 satisfies this equation
(B) x2 - x - 2 =0. None of 1 or 5 satisfies this equation
(C) 2x2 - 2 =0. x = 1 satisfies this equation. Hence this has a common root
(D) x2 - 2x - 3 =0. None of 1 or 5 satisfies this equation
(E) x^2 - 10x - 5 = 0. None of 1 or 5 satisfies this equation

Correct Option: C
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Re: Which of the following equations has a root in common with x^2 - 6x + [#permalink]

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New post 24 Jan 2018, 14:45
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Lolaergasheva wrote:
Which of the following equations has a root in common with \(x^2 - 6x + 5 = 0\)?

A. \(x^2 + 1 = 0\)

B. \(x^2 - x - 2 = 0\)

C. \(2x^2 - 2 = 0\)

D. \(x^2 - 2x - 3 = 0\)

E. \(x^2 - 10x - 5 = 0\)


Step 1: Solve the given equation: x² – 6x + 5 = 0
This is a quadratic set equal to zero, so let's factor to get:
(x-1)(x-5)=0
So, we have two solutions (roots): x=1 or x=5

Step 2: Solve the other 5 equations to see which one has a root (solution) of x=1 or x=5

IMPORTANT: It appears that the only way to answer this question is to keep checking every single answer choice until we find that one that has a solution of either x=1 or x=5. Given this, where do you think the test-maker would hide the correct answer? In these situations, I always start at E and work my way up. Is the answer to these questions always E (or perhaps D)? No, but it's more likely that the correct answer is near the bottom.

Okay, E: x² – 2x – 3 =0
Factor to get: (x-3)(x+1)=0
So, x=3 or x=-1
No shared solutions (roots) so keep moving.

D: 2x² – 2 =0
Factor: 2(x² - 1) = 0
Keep factoring: 2(x+1)(x-1)=0
So, x=1 or x=-1

We have a common solution, so the correct answer must be D

Cheers,
Brent
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Re: Which of the following equations has a root in common with x^2 - 6x +   [#permalink] 24 Jan 2018, 14:45
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