find two positive consecutive integers such that the square of first, added three times the second is 24.

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.

Find two positive consecutive integers such that the square of the first, added three times the second, is 24.

Learn how to solve a quadratic equation to find two consecutive integers where the square of the first integer plus three times the second equals 24. This step-by-step solution explains key concepts such as quadratic equations, formulas, and theorems. Perfect for Grades 7-9 students.

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