Understanding the Problem
We need to find a positive whole number, let's call it "x", that satisfies the equation derived from the statement:
"Five times a positive whole number is 3 less than twice the square of the number."
Formulating the Equation
From the problem, we can express this relationship as:
- 5x = 2x^2 - 3
This equation states that five times the number (5x) equals twice the square of the number (2x^2) minus 3.
Rearranging the Equation
To solve for x, we need to rearrange the equation:
- 2x^2 - 5x - 3 = 0
This is a standard quadratic equation in the form of ax^2 + bx + c = 0.
Applying the Quadratic Formula
The quadratic formula is:
- x = [ -b ± sqrt(b^2 - 4ac) ] / 2a
For our equation:
- a = 2
- b = -5
- c = -3
We substitute these values into the quadratic formula.
Calculating the Discriminant
First, calculate the discriminant (b^2 - 4ac):
- (-5)^2 - 4(2)(-3) = 25 + 24 = 49
Since the discriminant is positive, we will have two real solutions.
Finding the Solutions
Now, substitute back into the quadratic formula:
- x = [ 5 ± sqrt(49) ] / (2 * 2)
- x = [ 5 ± 7 ] / 4
This gives us two potential solutions:
- x = (12) / 4 = 3
- x = (-2) / 4 = -0.5 (not a positive whole number)
Conclusion
Thus, the positive whole number that satisfies the condition is:
- x = 3
This is the solution to the problem.