Exercise 32

One value of the constant \( k \) for the function \( f(x) = kx^2 + 2x + k \) (the equation \( kx^2 + 2x + k = 0 \) ) to have only one root, is:

One value of the constant \( k \) for the function

\[ \begin{aligned} &f(x) = kx^2 + 2x + k \\[.5em] &\hspace{3em} (\text{ the equation } kx^2 + 2x + k = 0) \end{aligned} \]

to have only one root, is:

  1. \( -1 \)

  2. \( i \)

  3. \( -i \)

  4. \( 4 \)

  5. \( -4 \)

Try to solve it before checking the answer.

  1. \( -1 \)

Since it is a quadratic equation, \( ax^2 + bx + c = 0) has only one real root if the discriminant is null. That is, if it is satisfied that:

\[ b^2 – 4ac = 0 \]

In our case:

\[ a=k, \quad b = 2, \quad \text{ and } \quad c = k \]

Then,

\[ \begin{aligned} b^2 – 4ac = 0 &\Rightarrow 2^2 – 4kk = 0 \\[.5em] &\Rightarrow 4 – 4k^2 = 0 \\[.5em] &\Rightarrow 4k^2 = 4 \\[.5em] &\Rightarrow k^2 = 1 \\[.5em] &\Rightarrow k = \pm 1 \end{aligned} \]

From the two roots we take \( \boldsymbol{ k = -1 }\), because it is the one that appears among the 5 choices.

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