find the value of k for which the roots of equation X^2 - 8kx + 2k = 0 has equal roots.
Answers
Answer:
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Step-by-step explanation:
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Answer:
Answer:
k = 0 , 1/8 .
Note:
• The possible values of x which satisfy the equation(in variable x ) are called its roots .
• A quadratic equation can have at most two roots.
• The discriminantnant D of a quadratic equation ,
ax² + bx + c = 0 , is given by;
D = b² - 4ac .
• If D = 0 , then both the roots of the quadratic equation are real and equal.
• If D > 0 , then both the roots of the quadratic equation are real and distinct.
• If D < 0 , then both the roots of the quadratic equation are imaginary.
Solution:
The given quadratic equation is;
x² - 8kx + 2k = 0.
Also,
The discriminant of the given quadratic equation will be;
=> D = (-8k)² - 4•1•2k
=> D = 64k² - 8k
=> D = 8k(8k - 1)
It is given that ,
The given quadratic equation has equal roots , hence its discriminant must be zero.
ie ;
=> D = 0
=> 8k(8k - 1) = 0
=> k(8k - 1) = 0
=> k = 0 OR (8k - 1) = 0
=> k = 0 OR 8k = 1
=> k = 0 OR k = 1/8
Hence,
The required values of k are :
0 and 1/8 .
Moreover;
Case1 : When k = 0
Then , the given quadratic equation will reduce to ;
=> x² - 8kx + 2k = 0
=> x² - 8•0•x + 2•0 = 0
=> x² = 0
Also,
Roots of the quadratic equation will be ;
x = 0 , 0
Case2 : When k = 4
Then , the given quadratic equation will reduce to ;
=> x² - 8kx + 2k = 0
=> x² - 8•(1/8)•x + 2•(1/8) = 0
=> x² - x + 1/4 = 0
=> x² - 2•x•(1/2) + (1/2)² = 0
=> (x - 1/2)² = 0
Also,
Roots of the quadratic equation will be ;
x = 1/2 , 1/2
Hence,
If k = 0 , then the given quadratic equation will reduce to x² = 0 and its equal roots will be 0 , 0 .
If k = 1/8 , then the given quadratic equation will reduce to (x - 1/2)² = 0 and its equal zeros will be 1/2 , 1/2 .
Step-by-step explanation:
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