if the square of a postive number is decreased by five times the number , the result is 14. Find the number
x^2 - 5x = 14
x^2 - 5x - 14 = 0
(x-7)(x+2) = 0
. . .
To find the number, let's set up an equation based on the given information.
Let's assume the number is "x". The square of the number is x^2.
According to the problem, the square of the number decreased by five times the number is equal to 14. Mathematically, this can be represented as:
x^2 - 5x = 14
To solve this equation, we need to bring all terms to one side and set it equal to zero:
x^2 - 5x - 14 = 0
Now, we have a quadratic equation. We can either factor it if possible or use the quadratic formula to find the values for x.
If we try factoring, we look for two numbers that multiply to -14 (the product of the last term and the constant) and add up to -5 (the coefficient of the middle term). In this case, there are no two numbers that fit these conditions. Hence, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 1 (coefficient of x^2), b = -5 (coefficient of x), and c = -14 (constant term).
Substituting these values into the quadratic formula:
x = (-(-5) ± √((-5)^2 - 4*1*(-14))) / 2*1
x = (5 ± √(25 + 56)) / 2
x = (5 ± √81) / 2
Now, we simplify further:
x = (5 ± 9) / 2
This gives us two possible solutions:
x1 = (5 + 9) / 2 = 14 / 2 = 7
x2 = (5 - 9) / 2 = -4 / 2 = -2
Since the problem states that the number is positive, the only valid solution is x = 7. Therefore, the number is 7.