The sum of a number and its square is 72. What is the number?
I took a guess at 8, but I don't know how to put it in an equation form.
n + n^2 = 72
If x = the number, then
x+x^2 = 72.
Solve for x.
To solve this problem, let's first express the problem in mathematical terms.
Let's call the number we are looking for "x". The sum of a number and its square can be expressed as "x + x^2".
According to the problem, "x + x^2 = 72".
Now, we can rearrange the equation to solve for "x". Start by subtracting 72 from both sides of the equation:
x + x^2 - 72 = 0
Next, let's set the equation equal to zero, as that makes it easier to work with:
x^2 + x - 72 = 0
Now, we have a quadratic equation in standard form. To solve it, we can either factor it or use the quadratic formula.
In this case, factoring is not straightforward, so let's use the quadratic formula:
For an equation in the form ax^2 + bx + c = 0, the quadratic formula states that the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = 1, and c = -72. Substituting these values into the quadratic formula:
x = (-1 ± √(1^2 - 4(1)(-72))) / (2(1))
Simplifying:
x = (-1 ± √(1 + 288)) / 2
x = (-1 ± √289) / 2
Now, we can simplify further:
x = (-1 ± 17) / 2
This yields two possible solutions:
1. x = (-1 + 17) / 2 = 16 / 2 = 8
2. x = (-1 - 17) / 2 = -18 / 2 = -9
Therefore, there are two possible numbers that satisfy the given equation: 8 and -9.