## Task A-4.2.

Numbers $a$ and $b$ are randomly chosen from the interval $[0,1]$. What is the probability that the equation $a x^{2}+b x+a=0$ has no real solutions?
## Solution.

The equation has no real solutions if and only if the discriminant is negative, i.e., $b^{2}<4 a^{2}$. Since $a, b \geqslant 0, b^{2}<4 a^{2}$ is equivalent to the condition $b<2 a$.

The desired probability geometrically corresponds to the ratio of the area under the line $b=2 a$ within the unit square to the area of the square itself.

![](https://cdn.mathpix.com/cropped/2024_05_30_0bfcef02cf5258b8afa5g-22.jpg?height=660&width=649&top_left_y=518&top_left_x=635)

The line $b=2 a$ intersects the line $b=1$ at the point $\left(\frac{1}{2}, 1\right)$, so the shaded region is a trapezoid with bases of length $1$ and $\frac{1}{2}$, and height of length $1$.

The probability that the equation $a x^{2}+b x+a=0$ has no solutions is $\frac{3}{4}$.