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If x 2 -hx-21=0,x 2 -3hx+35=0(h>0)has a common root ,then value of h is equal to A 1. B 2. C 3. D 4.

If x2-hx-21=0,x2-3hx+35=0(h>0)has a common root ,then value of h is equal toΒ 
A 1. Β B Β 2. Β  C 3. D 4.

Grade:11

1 Answers

Arun
25750 Points
5 years ago
β–Ί Let π‘Ž be the common roots of:Β 
Β Β Β π‘₯Β² – β„Žπ‘₯ – 21 = 0Β 
andΒ 
Β Β Β π‘₯Β² – 3β„Žπ‘₯ + 35 = 0Β 

Then by definition of a root, we have the system:Β 
Β Β Β { Β π‘ŽΒ² – β„Žπ‘Ž – 21 = 0Β 
Β Β Β { Β π‘ŽΒ² – 3β„Žπ‘Ž + 35 = 0Β 

β–Ί Subtracting second equation from the first yields:Β 
Β Β Β (π‘ŽΒ² – β„Žπ‘Ž – 21) – (π‘ŽΒ² – 3β„Žπ‘Ž + 35) = 0 – 0Β 
Β Β Β π‘ŽΒ² – β„Žπ‘Ž – 21 – π‘ŽΒ² + 3β„Žπ‘Ž – 35 = 0Β 
Β Β Β π‘ŽΒ² – π‘ŽΒ² – β„Žπ‘Ž + 3β„Žπ‘Ž – 21 – 35 = 0Β 
Β Β Β 0 + 2β„Žπ‘Ž – 56 = 0Β 
Β Β Β 2β„Žπ‘Ž = 56Β 
Β Β Β π‘Ž = 28/β„ŽΒ 

β–Ί Inserting this in the first equation:Β 
Β Β Β π‘ŽΒ² – β„Žπ‘Ž – 21 = 0Β 
Β Β Β (28/β„Ž)Β² – β„Ž(28/β„Ž) – 21 = 0    ← Since π‘Ž=28/β„ŽΒ 
Β Β Β 784/β„ŽΒ² – 28 – 21 = 0Β 
Β Β Β 784/β„ŽΒ² – 49 = 0Β 
Β Β Β 784/β„ŽΒ² = 49Β 
Β Β Β 16/β„ŽΒ² = 1 Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  Β  ← Divide both sides by 49Β 
Β Β Β Β 16 = β„ŽΒ² Β  Β  Β  Β  Β  Β Β  Β  Β  Β  Β  Β  Β  Β  Β  ← Multiply both sides by β„ŽΒ²Β 
Β Β Β Β β„Ž = Β±4Β 

β–Ί Now since we are told that β„Ž>0, the solution β„Ž=-4 is discarded leading to the final conclusion that:Β 
Β Β Β β„Ž = 4 Β  Β  Β  Β β—„ANSWER

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