Answer :
To solve the problem where one number is the square of another, and their sum is 182, follow these steps:
1. Define the numbers:
Let's call the first number [tex]\( x \)[/tex].
Then, the second number, which is the square of the first number, will be [tex]\( x^2 \)[/tex].
2. Set up the equation:
According to the problem, their sum is 182. So, we write the equation:
[tex]\[ x + x^2 = 182 \][/tex]
3. Rearrange the equation:
Rearrange the equation to bring all terms to one side:
[tex]\[ x^2 + x - 182 = 0 \][/tex]
4. Solve the quadratic equation:
Now, solve the quadratic equation [tex]\( x^2 + x - 182 = 0 \)[/tex].
Use the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -182 \)[/tex].
5. Calculate the discriminant:
[tex]\[ b^2 - 4ac = 1^2 - 4 \times 1 \times (-182) = 1 + 728 = 729 \][/tex]
6. Find the roots:
Since the discriminant is a perfect square, [tex]\( \sqrt{729} = 27 \)[/tex], the roots are:
[tex]\[ x = \frac{{-1 \pm 27}}{2} \][/tex]
7. Solve for [tex]\( x \)[/tex]:
[tex]\[ x_1 = \frac{{-1 + 27}}{2} = \frac{26}{2} = 13 \][/tex]
[tex]\[ x_2 = \frac{{-1 - 27}}{2} = \frac{-28}{2} = -14 \][/tex]
8. Determine [tex]\( y \)[/tex]:
If [tex]\( x = 13 \)[/tex], then [tex]\( y = x^2 = 13^2 = 169 \)[/tex].
If [tex]\( x = -14 \)[/tex], then [tex]\( y = x^2 = (-14)^2 = 196 \)[/tex].
Thus, the two possible pairs of numbers are:
- [tex]\( x = 13 \)[/tex] and [tex]\( y = 169 \)[/tex]
- [tex]\( x = -14 \)[/tex] and [tex]\( y = 196 \)[/tex]
These pairs satisfy the condition that one number is the square of the other, and their sum is 182.
1. Define the numbers:
Let's call the first number [tex]\( x \)[/tex].
Then, the second number, which is the square of the first number, will be [tex]\( x^2 \)[/tex].
2. Set up the equation:
According to the problem, their sum is 182. So, we write the equation:
[tex]\[ x + x^2 = 182 \][/tex]
3. Rearrange the equation:
Rearrange the equation to bring all terms to one side:
[tex]\[ x^2 + x - 182 = 0 \][/tex]
4. Solve the quadratic equation:
Now, solve the quadratic equation [tex]\( x^2 + x - 182 = 0 \)[/tex].
Use the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -182 \)[/tex].
5. Calculate the discriminant:
[tex]\[ b^2 - 4ac = 1^2 - 4 \times 1 \times (-182) = 1 + 728 = 729 \][/tex]
6. Find the roots:
Since the discriminant is a perfect square, [tex]\( \sqrt{729} = 27 \)[/tex], the roots are:
[tex]\[ x = \frac{{-1 \pm 27}}{2} \][/tex]
7. Solve for [tex]\( x \)[/tex]:
[tex]\[ x_1 = \frac{{-1 + 27}}{2} = \frac{26}{2} = 13 \][/tex]
[tex]\[ x_2 = \frac{{-1 - 27}}{2} = \frac{-28}{2} = -14 \][/tex]
8. Determine [tex]\( y \)[/tex]:
If [tex]\( x = 13 \)[/tex], then [tex]\( y = x^2 = 13^2 = 169 \)[/tex].
If [tex]\( x = -14 \)[/tex], then [tex]\( y = x^2 = (-14)^2 = 196 \)[/tex].
Thus, the two possible pairs of numbers are:
- [tex]\( x = 13 \)[/tex] and [tex]\( y = 169 \)[/tex]
- [tex]\( x = -14 \)[/tex] and [tex]\( y = 196 \)[/tex]
These pairs satisfy the condition that one number is the square of the other, and their sum is 182.
