Question

Find the value of k if x+1/2 is one of the factors of the polynomial x^2 +2x + k

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Answer 1

Given that,

$\sf{x+ \dfrac{1}{2}\ is\ a\ factor\ of\ x^2+2x+k.}$

To find :-

The value of k.

Solution :-

→ Finding the zero of x + 1/2 :-

$\displaystyle{\sf{x+\frac{1}{2}=0 }}\\\\\displaystyle{\sf{\implies x=-\frac{1}{2}}}$

◙ Let p(x) = $\sf{x^2+2x+k}$.

A/q,

$\displaystyle{\sf{p(-\frac{1}{2} )=0}}$

Substituting the value of x as -1/2 :-

$\displaystyle{\sf{\implies (- \frac{1}{2})^2+2(-\frac{1}{2})+k=0}}\\\\\displaystyle{\sf{\implies \frac{1}{4}-1+k=0}}\\\\\displaystyle{\sf{\implies \frac{1-4}{4}+k=0 }}\\\\\displaystyle{\sf{\implies \frac{-3}{4}+k=0}}\\\\\boxed{\displaystyle{\sf{\implies k=\frac{3}{4} }}}$

Hence, for the value of k for which (x + 1/2) is a factor of x² + 2x + k, is 3/4.

Answer 2

Given :

• Polynomial is x² + 2x - k
• One factor of equation is x + 1/2 = 0

To Find :

• Value of k

Solution :

we're given that the quadratic polynomial is x² + 2x - k. And one of its factor is x + 1/2. So,

⇒x + 1/2 = 0

⇒x = -1/2

$\therefore$ Value of x is -1/2

________________________________

Put value of x in the polynomial

⇒x² + 2x + k = 0

⇒(-1/2)² + 2(-1/2) + k = 0

⇒1/4 + (2/-2) + k = 0

⇒1/4 - 1 + k = 0

⇒(1 - 4)/4 + k = 0

⇒-3/4 + k = 0

⇒k = 3/4

$\therefore$ Value of k is 3/4