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If a ,b ,c are real numbers such that ac does not equal to 0 , then show that at least one of the equations ax^2+ bx+ c =0 and -ax^2+ bx +c=0 has real roots.?
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Proof:
To prove that at least one of the equations ax^2 + bx + c = 0 and -ax^2 + bx + c = 0 has real roots, we will consider two cases:
Case 1: When a > 0
In this case, the equation ax^2 + bx + c = 0 represents a quadratic function with a positive leading coefficient. The graph of this function will open upwards.
Case 1.1: Discriminant D ≥ 0
If the discriminant D = b^2 - 4ac ≥ 0, then the quadratic equation ax^2 + bx + c = 0 will have real roots. This is because the discriminant determines the nature of the roots:
- D > 0: Two distinct real roots
- D = 0: One real root (repeated)
- D < 0:="" no="" real="" roots="" />
Case 1.2: Discriminant D < />
If the discriminant D = b^2 - 4ac < 0,="" then="" the="" quadratic="" equation="" ax^2="" +="" bx="" +="" c="0" will="" not="" have="" real="" roots.="" however,="" in="" this="" case,="" we="" can="" consider="" the="" equation="" -ax^2="" +="" bx="" +="" c="0." />
Since a > 0, the equation -ax^2 + bx + c = 0 represents a quadratic function with a negative leading coefficient. The graph of this function will open downwards.
Case 1.2.1: Discriminant D' ≥ 0
If the discriminant D' = b^2 - 4(-a)c = b^2 + 4ac ≥ 0, then the quadratic equation -ax^2 + bx + c = 0 will have real roots.
Case 1.2.2: Discriminant D' < />
If the discriminant D' = b^2 + 4ac < 0,="" then="" the="" quadratic="" equation="" -ax^2="" +="" bx="" +="" c="0" will="" not="" have="" real="" roots.="" />
Case 2: When a < />
In this case, the equation ax^2 + bx + c = 0 represents a quadratic function with a negative leading coefficient. The graph of this function will open downwards.
Case 2.1: Discriminant D ≥ 0
If the discriminant D = b^2 - 4ac ≥ 0, then the quadratic equation ax^2 + bx + c = 0 will have real roots.
Case 2.2: Discriminant D < />
If the discriminant D = b^2 - 4ac < 0,="" then="" the="" quadratic="" equation="" ax^2="" +="" bx="" +="" c="0" will="" not="" have="" real="" roots.="" however,="" in="" this="" case,="" we="" can="" consider="" the="" equation="" -ax^2="" +="" bx="" +="" c="0." />
Since a < 0,="" the="" equation="" -ax^2="" +="" bx="" +="" c="0" represents="" a="" quadratic="" function="" with="" a="" positive="" leading="" coefficient.="" the="" graph="" of="" this="" function="" will="" open="" upwards.="" />
Case 2.2.1: Discriminant D' ≥ 0
If the discriminant D' = b^2 - 4(-
To prove that at least one of the equations ax^2 + bx + c = 0 and -ax^2 + bx + c = 0 has real roots, we will consider two cases:
Case 1: When a > 0
In this case, the equation ax^2 + bx + c = 0 represents a quadratic function with a positive leading coefficient. The graph of this function will open upwards.
Case 1.1: Discriminant D ≥ 0
If the discriminant D = b^2 - 4ac ≥ 0, then the quadratic equation ax^2 + bx + c = 0 will have real roots. This is because the discriminant determines the nature of the roots:
- D > 0: Two distinct real roots
- D = 0: One real root (repeated)
- D < 0:="" no="" real="" roots="" />
Case 1.2: Discriminant D < />
If the discriminant D = b^2 - 4ac < 0,="" then="" the="" quadratic="" equation="" ax^2="" +="" bx="" +="" c="0" will="" not="" have="" real="" roots.="" however,="" in="" this="" case,="" we="" can="" consider="" the="" equation="" -ax^2="" +="" bx="" +="" c="0." />
Since a > 0, the equation -ax^2 + bx + c = 0 represents a quadratic function with a negative leading coefficient. The graph of this function will open downwards.
Case 1.2.1: Discriminant D' ≥ 0
If the discriminant D' = b^2 - 4(-a)c = b^2 + 4ac ≥ 0, then the quadratic equation -ax^2 + bx + c = 0 will have real roots.
Case 1.2.2: Discriminant D' < />
If the discriminant D' = b^2 + 4ac < 0,="" then="" the="" quadratic="" equation="" -ax^2="" +="" bx="" +="" c="0" will="" not="" have="" real="" roots.="" />
Case 2: When a < />
In this case, the equation ax^2 + bx + c = 0 represents a quadratic function with a negative leading coefficient. The graph of this function will open downwards.
Case 2.1: Discriminant D ≥ 0
If the discriminant D = b^2 - 4ac ≥ 0, then the quadratic equation ax^2 + bx + c = 0 will have real roots.
Case 2.2: Discriminant D < />
If the discriminant D = b^2 - 4ac < 0,="" then="" the="" quadratic="" equation="" ax^2="" +="" bx="" +="" c="0" will="" not="" have="" real="" roots.="" however,="" in="" this="" case,="" we="" can="" consider="" the="" equation="" -ax^2="" +="" bx="" +="" c="0." />
Since a < 0,="" the="" equation="" -ax^2="" +="" bx="" +="" c="0" represents="" a="" quadratic="" function="" with="" a="" positive="" leading="" coefficient.="" the="" graph="" of="" this="" function="" will="" open="" upwards.="" />
Case 2.2.1: Discriminant D' ≥ 0
If the discriminant D' = b^2 - 4(-
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If a ,b ,c are real numbers such that ac does not equal to 0 , then show that at least one of the equations ax^2+ bx+ c =0 and -ax^2+ bx +c=0 has real roots.?
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If a ,b ,c are real numbers such that ac does not equal to 0 , then show that at least one of the equations ax^2+ bx+ c =0 and -ax^2+ bx +c=0 has 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 a ,b ,c are real numbers such that ac does not equal to 0 , then show that at least one of the equations ax^2+ bx+ c =0 and -ax^2+ bx +c=0 has real roots.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a ,b ,c are real numbers such that ac does not equal to 0 , then show that at least one of the equations ax^2+ bx+ c =0 and -ax^2+ bx +c=0 has real roots.?.
If a ,b ,c are real numbers such that ac does not equal to 0 , then show that at least one of the equations ax^2+ bx+ c =0 and -ax^2+ bx +c=0 has 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 a ,b ,c are real numbers such that ac does not equal to 0 , then show that at least one of the equations ax^2+ bx+ c =0 and -ax^2+ bx +c=0 has real roots.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a ,b ,c are real numbers such that ac does not equal to 0 , then show that at least one of the equations ax^2+ bx+ c =0 and -ax^2+ bx +c=0 has real roots.?.
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