Eight times the square of a positive integer divided by two equals 144 minus 36 times the positive integer. Find the unknown positive integer.

A) 2
B) 3
C) 5
D) 7

Question

High School

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Answer :

DanielJosephParker DanielJosephParker · Answer 1 · 2025-02-20T11:24:52-05:00

Let's break down the problem step-by-step.

We are given a mathematical equation as follows:

Eight times the square of a positive integer divided by two equals 144 minus 36 times the positive integer.

Let the positive integer be [tex]x[/tex].
The equation based on the description is:
[tex]\frac{8x^2}{2} = 144 - 36x[/tex]
Simplify the left side of the equation:
[tex]4x^2 = 144 - 36x[/tex]
Rearrange the equation to bring all terms to one side:
[tex]4x^2 + 36x - 144 = 0[/tex]
To simplify solving, divide the entire equation by 4:
[tex]x^2 + 9x - 36 = 0[/tex]
Now, we will solve the quadratic equation using the quadratic formula:
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
where [tex]a = 1[/tex], [tex]b = 9[/tex], and [tex]c = -36[/tex].
Calculate the discriminant:
[tex]b^2 - 4ac = 9^2 - 4 \times 1 \times (-36) = 81 + 144 = 225[/tex]
Since the discriminant is positive, we have two real solutions:
[tex]x = \frac{-9 \pm \sqrt{225}}{2}[/tex]
Calculate the solutions:
[tex]x = \frac{-9 + 15}{2} = \frac{6}{2} = 3[/tex]
[tex]x = \frac{-9 - 15}{2} = \frac{-24}{2} = -12[/tex]
Since we need the positive integer, we choose [tex]x = 3[/tex].

Therefore, the unknown positive integer is 3.

The correct option is B) 3.