Consider the quadratic equation $x ^ { 2 } + b x + c = 0$, where $b$ and $c$ are chosen randomly from the interval $[ 0,1 ]$ with the probability uniformly distributed over all pairs $( b , c )$. Let $p ( b ) =$ the probability that the given equation has a real solution for given (fixed) value of $b$. Answer the following questions by filling in the blanks. a) The equation $x ^ { 2 } + b x + c = 0$ has a real solution if and only if $b ^ { 2 } - 4 c$ is Answer: $\_\_\_\_$ b) The value of $p \left( \frac { 1 } { 2 } \right)$, i.e., the probability that $x ^ { 2 } + \frac { x } { 2 } + c = 0$ has a real solution is Answer: $\_\_\_\_$ c) As a function of $b$, is $p ( b )$ increasing, decreasing or constant? Answer: $\_\_\_\_$ d) As $b$ and $c$ both vary, what is the probability that $x ^ { 2 } + b x + c = 0$ has a real solution? Answer: $\_\_\_\_$
Consider the quadratic equation $x ^ { 2 } + b x + c = 0$, where $b$ and $c$ are chosen randomly from the interval $[ 0,1 ]$ with the probability uniformly distributed over all pairs $( b , c )$. Let $p ( b ) =$ the probability that the given equation has a real solution for given (fixed) value of $b$. Answer the following questions by filling in the blanks.\\
a) The equation $x ^ { 2 } + b x + c = 0$ has a real solution if and only if $b ^ { 2 } - 4 c$ is
Answer: $\_\_\_\_$\\
b) The value of $p \left( \frac { 1 } { 2 } \right)$, i.e., the probability that $x ^ { 2 } + \frac { x } { 2 } + c = 0$ has a real solution is
Answer: $\_\_\_\_$\\
c) As a function of $b$, is $p ( b )$ increasing, decreasing or constant?
Answer: $\_\_\_\_$\\
d) As $b$ and $c$ both vary, what is the probability that $x ^ { 2 } + b x + c = 0$ has a real solution?
Answer: $\_\_\_\_$