Write All the Values of k for Which x² + kx + 16 = 0 Has Equal Roots. Find the Roots of the Equation So Obtained.

Question:

\( x^2 + kx + 16 = 0 \)

Write all the values of \(k\) for which the equation has equal roots. Find the roots of the equation so obtained.

Solution

For equal roots, the discriminant must be zero.

\( D=b^2-4ac=0 \)

Here,

\( a=1,\quad b=k,\quad c=16 \)

Therefore,

\( k^2-4(1)(16)=0 \)

\( k^2-64=0 \)

\( (k-8)(k+8)=0 \)

\( k=8 \quad \text{or} \quad k=-8 \)

When \(k=8\):

\( x^2+8x+16=0 \)

\( (x+4)^2=0 \)

\( x=-4,\,-4 \)

When \(k=-8\):

\( x^2-8x+16=0 \)

\( (x-4)^2=0 \)

\( x=4,\,4 \)

\( \boxed{k=8 \text{ or } k=-8} \)

For \(k=8\), equal roots are \( -4,\,-4 \)

For \(k=-8\), equal roots are \( 4,\,4 \)

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