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The quadratic equation

x^{2} + (k + 1) x + 2 k = 0

does not have distinct roots.

Find the range of possible value of $k .$

Use the table below to find your answer based on your solution set.

Give your answer to $3$ significant figures.

Hint

The wording implies that there is either $1$ or $0$ roots.

Hint

b^{2} - 4 a c \leq 0

Solution

The wording implies that there is either $1$ or $0$ roots.

This means the discriminant satisfies

\begin{aligned} b^{2} - 4 a c & \leq 0 \\ (k + 1)^{2} - 4 (1) (2 k) & \leq 0 \\ k^{2} + 2 k + 1 - 8 k & \leq 0 \\ k^{2} - 6 k + 1 & \leq 0 \end{aligned}

By the quadratic formula, we find the roots of $k^{2} - 6 k + 1$ are

\begin{aligned} k & = 3 \pm 2 \sqrt{2} \end{aligned}

and so we make a sketch

parabola

From here we see that the solution is

3 - 2 \sqrt{2} \leq k \leq 3 + 2 \sqrt{2}