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The number of possible integral values for y for which (y-2)x^2+8x+(y+4)>0 for all x belongs to R is (answer is 7) ?
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The number of possible integral values for y for which (y-2)x^2+8x+(y+...
Analysis of the Inequality:
The given inequality is in the form of a quadratic expression which must be greater than zero for all real values of x. Let's analyze the inequality step by step to find the possible integral values of y.
Quadratic Inequality:
(y-2)x^2 + 8x + (y+4) > 0
Discriminant:
For the quadratic expression to be always positive, the discriminant must be less than zero since the coefficient of x^2 is positive.
Discriminant Calculation:
8^2 - 4(y-2)(y+4) < />
64 - 4(y^2 + 2y - 8) < />
64 - 4y^2 - 8y + 32 < />
-4y^2 - 8y + 32 < />
-4y^2 - 8y + 96 < />
y^2 + 2y - 24 > 0
Factorization:
(y+6)(y-4) > 0
Sign Analysis:
The inequality is satisfied when y < -6="" or="" y="" /> 4. Therefore, the possible integral values for y are y ∈ {..., -8, -7, -6, 5, 6, 7, ...}.
The given inequality is in the form of a quadratic expression which must be greater than zero for all real values of x. Let's analyze the inequality step by step to find the possible integral values of y.
Quadratic Inequality:
(y-2)x^2 + 8x + (y+4) > 0
Discriminant:
For the quadratic expression to be always positive, the discriminant must be less than zero since the coefficient of x^2 is positive.
Discriminant Calculation:
8^2 - 4(y-2)(y+4) < />
64 - 4(y^2 + 2y - 8) < />
64 - 4y^2 - 8y + 32 < />
-4y^2 - 8y + 32 < />
-4y^2 - 8y + 96 < />
y^2 + 2y - 24 > 0
Factorization:
(y+6)(y-4) > 0
Sign Analysis:
The inequality is satisfied when y < -6="" or="" y="" /> 4. Therefore, the possible integral values for y are y ∈ {..., -8, -7, -6, 5, 6, 7, ...}.
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The number of possible integral values for y for which (y-2)x^2+8x+(y+4)>0 for all x belongs to R is (answer is 7) ?
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The number of possible integral values for y for which (y-2)x^2+8x+(y+4)>0 for all x belongs to R is (answer is 7) ? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The number of possible integral values for y for which (y-2)x^2+8x+(y+4)>0 for all x belongs to R is (answer is 7) ? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of possible integral values for y for which (y-2)x^2+8x+(y+4)>0 for all x belongs to R is (answer is 7) ?.
The number of possible integral values for y for which (y-2)x^2+8x+(y+4)>0 for all x belongs to R is (answer is 7) ? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The number of possible integral values for y for which (y-2)x^2+8x+(y+4)>0 for all x belongs to R is (answer is 7) ? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of possible integral values for y for which (y-2)x^2+8x+(y+4)>0 for all x belongs to R is (answer is 7) ?.
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