Let's define the two consecutive integers as $x$ and $x + 1$.

According to the problem, the square of the first integer, added to three times the second integer, equals 24. This can be expressed as the following equation:

$x^2 + 3(x + 1) = 24$

Step 1: Expand and simplify the equation

$x^2 + 3(x + 1) = 24$ $x^2 + 3x + 3 = 24$

Step 2: Subtract 24 from both sides to set the equation equal to zero:

$x^2 + 3x + 3 - 24 = 0$ $x^2 + 3x - 21 = 0$

Step 3: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 1$, $b = 3$, and $c = -21$.

Substitute these values into the quadratic formula:

$x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-21)}}{2(1)}$ $x = \frac{-3 \pm \sqrt{9 + 84}}{2}$ $x = \frac{-3 \pm \sqrt{93}}{2}$

Step 4: Approximate the square root

The square root of 93 is approximately $9.64$, so:

$x = \frac{-3 \pm 9.64}{2}$

Step 5: Solve for $x$

1. $x = \frac{-3 + 9.64}{2} = \frac{6.64}{2} = 3.32$
2. $x = \frac{-3 - 9.64}{2} = \frac{-12.64}{2} = -6.32$

Since we are looking for positive integers, we discard these values because $x$ is not an integer. There is no integer solution to this quadratic equation that meets the requirements.

Thus, the problem as stated does not have a solution for positive consecutive integers.

Do you want more details or have any questions? Here are some related problems for practice:

1. Find two consecutive integers whose product is 72.
2. Find two positive consecutive integers such that their sum is 25.
3. Solve $x^2 + 5x + 6 = 0$.
4. Find two consecutive odd integers whose product is 99.
5. Find two consecutive even integers whose sum is 50.

Tip: When solving quadratic equations, always check if the discriminant is a perfect square for integer solutions.