How do you solve this? Find the values for $c$ so that $x^{2} + 2 x + c = 0$ will have two solutions.

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Solution

Find the value of $c$ :
It is given the following quadratic equation :
$x^{2} + 2 x + c = 0$ .
By compare with general equation $A x^{2} + B x + C = 0$ :
$A = 1, B = 2, C = c$
It is known that a quadratic equation has two solutions if the discriminant is greater than zero.
Then :
$B^{2} - 4 A C \geq 0$
This gives :
$\begin{array}{rcl} 2^{2} - 4 c & \geq & 0 \\ \Rightarrow & 4 - 4 c \geq 0 \\ \Rightarrow & 4 \geq 4 c \\ \Rightarrow & 1 \geq c \\ \Rightarrow & c \leq 1 \end{array}$
Therefore, the required values of $c$ are $c \leq 1$ .

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