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If ad does not equal bc then prove that equation (a^2 b^2)x^2 (ac bd)x (c^2 d^2 )=0 Has no real roots?
Most Upvoted Answer
If ad does not equal bc then prove that equation (a^2 b^2)x^2 (ac b...
Proof:
To prove that the equation has no real roots, we can consider the discriminant of the quadratic equation. The discriminant is given by the expression b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.
In this case, the quadratic equation is (a^2 b^2)x^2 (ac bd)x (c^2 d^2 )=0.
Discriminant:
The discriminant of the given quadratic equation is (ac bd)^2 - 4(a^2 b^2)(c^2 d^2 ).
Case 1: ad ≠ bc
If ad ≠ bc, then the discriminant is given by (ac bd)^2 - 4(a^2 b^2)(c^2 d^2 ).
Subcase 1.1: ac bd ≠ 0
If ac bd ≠ 0, then the discriminant is (ac bd)^2 - 4(a^2 b^2)(c^2 d^2 ).
Subcase 1.1.1: (ac bd)^2 > 4(a^2 b^2)(c^2 d^2 )
If (ac bd)^2 > 4(a^2 b^2)(c^2 d^2 ), then the discriminant is positive.
In this case, the quadratic equation has two distinct real roots.
Subcase 1.1.2: (ac bd)^2 < 4(a^2="" b^2)(c^2="" d^2="" />
If (ac bd)^2 < 4(a^2="" b^2)(c^2="" d^2="" ),="" then="" the="" discriminant="" is="" />
In this case, the quadratic equation has two complex roots, which means it has no real roots.
Subcase 1.2: ac bd = 0
If ac bd = 0, then the discriminant is 0 - 4(a^2 b^2)(c^2 d^2 ).
In this case, the discriminant is negative or zero, which means the quadratic equation has no real roots.
Case 2: ad = bc
If ad = bc, then the discriminant is (ac bd)^2 - 4(a^2 b^2)(c^2 d^2 ).
Subcase 2.1: ac bd ≠ 0
If ac bd ≠ 0, then the discriminant is (ac bd)^2 - 4(a^2 b^2)(c^2 d^2 ).
In this case, the discriminant is zero, which means the quadratic equation has one real root.
Subcase 2.2: ac bd = 0
If ac bd = 0, then the discriminant is 0 - 4(a^2 b^2)(c^2 d^2 ).
In this case, the discriminant is negative or zero, which means the quadratic equation has no real roots.
Conclusion:
Therefore, regardless of whether ad ≠ bc or ad = bc, the quadratic equation (a^2 b^2)x^2 (ac bd)x
To prove that the equation has no real roots, we can consider the discriminant of the quadratic equation. The discriminant is given by the expression b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.
In this case, the quadratic equation is (a^2 b^2)x^2 (ac bd)x (c^2 d^2 )=0.
Discriminant:
The discriminant of the given quadratic equation is (ac bd)^2 - 4(a^2 b^2)(c^2 d^2 ).
Case 1: ad ≠ bc
If ad ≠ bc, then the discriminant is given by (ac bd)^2 - 4(a^2 b^2)(c^2 d^2 ).
Subcase 1.1: ac bd ≠ 0
If ac bd ≠ 0, then the discriminant is (ac bd)^2 - 4(a^2 b^2)(c^2 d^2 ).
Subcase 1.1.1: (ac bd)^2 > 4(a^2 b^2)(c^2 d^2 )
If (ac bd)^2 > 4(a^2 b^2)(c^2 d^2 ), then the discriminant is positive.
In this case, the quadratic equation has two distinct real roots.
Subcase 1.1.2: (ac bd)^2 < 4(a^2="" b^2)(c^2="" d^2="" />
If (ac bd)^2 < 4(a^2="" b^2)(c^2="" d^2="" ),="" then="" the="" discriminant="" is="" />
In this case, the quadratic equation has two complex roots, which means it has no real roots.
Subcase 1.2: ac bd = 0
If ac bd = 0, then the discriminant is 0 - 4(a^2 b^2)(c^2 d^2 ).
In this case, the discriminant is negative or zero, which means the quadratic equation has no real roots.
Case 2: ad = bc
If ad = bc, then the discriminant is (ac bd)^2 - 4(a^2 b^2)(c^2 d^2 ).
Subcase 2.1: ac bd ≠ 0
If ac bd ≠ 0, then the discriminant is (ac bd)^2 - 4(a^2 b^2)(c^2 d^2 ).
In this case, the discriminant is zero, which means the quadratic equation has one real root.
Subcase 2.2: ac bd = 0
If ac bd = 0, then the discriminant is 0 - 4(a^2 b^2)(c^2 d^2 ).
In this case, the discriminant is negative or zero, which means the quadratic equation has no real roots.
Conclusion:
Therefore, regardless of whether ad ≠ bc or ad = bc, the quadratic equation (a^2 b^2)x^2 (ac bd)x
Community Answer
If ad does not equal bc then prove that equation (a^2 b^2)x^2 (ac b...
(a²+b²)x²+2(ac+bd)x+(c²+d²)=0
a=(a²+b²) b=2(ac+bd) c=(c²+d²)
putting a,b,c in formula
=> [-b +/- √(b²-4ac)]/2a
=> {-2(ac+bd) +/-√[4(ac+bd)²-4(a²+b²)(c²+d²)]}/2(a²+b²)
=> [-2(ac+bd)+/- √4a²c²+4b²d²+8abcd-4a²c²-4a²d²-4b²c²-4
b²d²]/2(a²+b²)
=> -2(ac+bd) +/- √ - (4a²d²-8abcd+4b²c²)
Hence root cant be negative.
So, it has no real roots.
That's all🙂
a=(a²+b²) b=2(ac+bd) c=(c²+d²)
putting a,b,c in formula
=> [-b +/- √(b²-4ac)]/2a
=> {-2(ac+bd) +/-√[4(ac+bd)²-4(a²+b²)(c²+d²)]}/2(a²+b²)
=> [-2(ac+bd)+/- √4a²c²+4b²d²+8abcd-4a²c²-4a²d²-4b²c²-4
b²d²]/2(a²+b²)
=> -2(ac+bd) +/- √ - (4a²d²-8abcd+4b²c²)
Hence root cant be negative.
So, it has no real roots.
That's all🙂
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If ad does not equal bc then prove that equation (a^2 b^2)x^2 (ac bd)x (c^2 d^2 )=0 Has no real roots?
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If ad does not equal bc then prove that equation (a^2 b^2)x^2 (ac bd)x (c^2 d^2 )=0 Has no real roots? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If ad does not equal bc then prove that equation (a^2 b^2)x^2 (ac bd)x (c^2 d^2 )=0 Has no real roots? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If ad does not equal bc then prove that equation (a^2 b^2)x^2 (ac bd)x (c^2 d^2 )=0 Has no real roots?.
If ad does not equal bc then prove that equation (a^2 b^2)x^2 (ac bd)x (c^2 d^2 )=0 Has no real roots? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If ad does not equal bc then prove that equation (a^2 b^2)x^2 (ac bd)x (c^2 d^2 )=0 Has no real roots? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If ad does not equal bc then prove that equation (a^2 b^2)x^2 (ac bd)x (c^2 d^2 )=0 Has no real roots?.
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