Question

Find the value of k so that the given has equal roots (k+1)x²+(7k+7)x+4K+9

Answer 1

Given :

• (k+1)x² + (7k+7)x + 4k+9
• Roots are equal.

To Find :

• Value of k.

Solution :

We know an polynomial has equal roots only if the discriminant,D equals to zero.

Compare the given polynomial with the general form.

General Form :

• ax² + bx + c

On comparing we get,

• a = (k+1)
• b = (7k + 7)
• c = 4k + 9

We know,

$\bold{D\:=\:b^2\:-\:4ac}$

Block in the data,

$\sf{\longrightarrow{0\:=\:(7k+7)^2-4\big[(k+1)(4k+9)}\big]}$

$\sf{\longrightarrow{0\:=\:(7k) ^2\:+\:(2\:\times\:7k\:\times\:7)\:+\:(7)^2\:-\:4\:\times\:\big[k(4k+9)+1(4k+9)\big]}}$

$\bold{\big[Using\:(a+b)^2\:=\:a^2\:+\:2ab\:+\:b^2\big]}$

$\sf{\longrightarrow{0\:=\:49k^2\:+\:(14k\:\times\:7\:)\times\:49\:-\:4\:\times\:\big[4k^2+9k\:+\:4k+9}\big]}$

$\sf{\longrightarrow{0\:=\:49k^2\:+\:98k\:+\:49\:-\:4\:\times\:\big[4k^2+13k+9}\big]}$

$\sf{\longrightarrow{0\:=\:49k^2+98k+49\:-\:16k^2-52k-36}}$

$\sf{\longrightarrow{0\:=\:49k^2-16k^2+98k-52k+49-36}}$

$\sf{\longrightarrow{33k^2+46k+13=0}}$

We can further solve the above equation by using factorization method.

$\sf{\longrightarrow{33k^2+33k+13k+13=0}}$

$\sf{\longrightarrow{33k(k+1)+13(k+1)=0}}$

$\sf{\longrightarrow{(k+1)(33k+13)=0}}$

$\sf{\longrightarrow{k+1=0\:\:or\:\:33k+13=0}}$

$\sf{\longrightarrow{k=-1\:\:or\:\:33k=-13}}$

$\sf{\longrightarrow{k=-1\:\:or\:\:k=\dfrac{-13}{33}}}$

$\large{\boxed{\bold{Value\:of\:k\:=\:-1\:or\:\dfrac{-13}{33}}}}$

Answer 2

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$\huge\tt{GIVEN:}$

• Polynomial (k+1)x²+(7k+7)x+4K+9 whose roots are equal

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$\huge\tt{TO~FIND:}$

• The value of K

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$\huge\tt{SOLUTION:}$

As we are known that an polynomial has equal roots only if the discriminant D , is zero

Now,

Comparing the polynomial with general form.. we get

• a = (k+1)
• b = (7k +7)
• c = 4k + 9

as we know that,

D = b² - 4ac

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↪0 = (7k+7²)-4[(k+1)(4k+9)]

↪0 = (7k²) + (2×7k×7) + (7²) - 4 × [k(4k+9)+1(4k+9)]

↪0 = 49k² + 98k + 49 - 16k² - 52k - 36

↪0 = 49k²-16k² + 98k - 52k + 49 - 36

↪0 = 33k² + 46k + 13

↪0 = 33k (k+1) + 13(k+1)

↪0 = 33k + 13

↪-13 = 33k

↪-13/33 = k

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$\tt\purple{Value\:of\:k\:=\:-1\:or\:\dfrac{-13}{33}}$

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