GMAT Exam > GMAT Questions > (x2 + 1)2 - x2 = 0 hasa)no real rootsb)4 real...
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(x2 + 1)2 - x2 = 0 has
- a)no real roots
- b)4 real roots
- c)2 real roots
- d)1 real root
Correct answer is option 'A'. Can you explain this answer?
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(x2 + 1)2 - x2 = 0 hasa)no real rootsb)4 real rootsc)2 real rootsd)1 r...
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(x2 + 1)2 - x2 = 0 hasa)no real rootsb)4 real rootsc)2 real rootsd)1 r...
**Solution:**
To solve the given equation, we can start by simplifying the expression on the left-hand side:
(x^2 + 1)^2 - x^2 = 0
Expanding the square of the binomial, we get:
(x^4 + 2x^2 + 1) - x^2 = 0
Combining like terms, we have:
x^4 + x^2 + 1 = 0
Now, let's substitute a variable to make it easier to solve. Let's set y = x^2. Substituting this into the equation, we get:
y^2 + y + 1 = 0
This is now a quadratic equation in terms of y. We can use the quadratic formula to solve for y:
y = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 1, and c = 1. Substituting these values into the formula, we get:
y = (-1 ± √(1 - 4(1)(1))) / 2(1)
Simplifying further:
y = (-1 ± √(1 - 4)) / 2
y = (-1 ± √(-3)) / 2
Since the discriminant (√(-3)) is negative, the quadratic equation has no real roots. Therefore, the original equation (x^2 + 1)^2 - x^2 = 0 also has no real roots.
Hence, the correct answer is option A) no real roots.
To solve the given equation, we can start by simplifying the expression on the left-hand side:
(x^2 + 1)^2 - x^2 = 0
Expanding the square of the binomial, we get:
(x^4 + 2x^2 + 1) - x^2 = 0
Combining like terms, we have:
x^4 + x^2 + 1 = 0
Now, let's substitute a variable to make it easier to solve. Let's set y = x^2. Substituting this into the equation, we get:
y^2 + y + 1 = 0
This is now a quadratic equation in terms of y. We can use the quadratic formula to solve for y:
y = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 1, and c = 1. Substituting these values into the formula, we get:
y = (-1 ± √(1 - 4(1)(1))) / 2(1)
Simplifying further:
y = (-1 ± √(1 - 4)) / 2
y = (-1 ± √(-3)) / 2
Since the discriminant (√(-3)) is negative, the quadratic equation has no real roots. Therefore, the original equation (x^2 + 1)^2 - x^2 = 0 also has no real roots.
Hence, the correct answer is option A) no real roots.
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Community Answer
(x2 + 1)2 - x2 = 0 hasa)no real rootsb)4 real rootsc)2 real rootsd)1 r...
Option (A)
Given,
(
x
2
+
1
)
2
−
x
2
=
0
⇒
(
x
2
+
1
)
2
=
x
2
Square root on both sides. We get,
x
2
+
1
=
x
o
r
x
2
+
1
=
−
x
⇒
x
2
−
x
+
1
=
0
o
r
x
2
+
x
+
1
=
0
discriminant for the above equations will be;
1-4 = -3 <0 or 1-4=-3 <0
Hence,
(
x
2
+
1
)
2
−
x
2
=
0
has no real roots.
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