Question

When 238 is divided by the natural number n, the quotient is 3 more than n, and there is no remainder. What is n?

Answer 1

Answer:

n = 14

Step-by-step explanation:

Write the equation in algebraic (numbers and symbols) form

$\frac{238}{n}$ = n + 3              Rearrange the equation

$n\frac{238}{n}$ = n(n + 3)           Multiply both sides by 'n'

238 = n(n + 3)          On the left, multiplying and dividing by 'n' cancels out

238 = n² + 3n           Distributed right side

0 = n² + 3n - 238        Rearranged to standard form for quadratic equations

Remember standard form for quadratic equations is: ax² + bx + c = 0

State the variables a, b and c:

a = 1; b = 3; c = -238

When a quadratic equation is in standard form, you can use the quadratic formula to solve for 'n'. (Usually, you see the variable 'x', but the quadratic formula can solve for 'n' as if it is 'x', as long as it's in standard form.)

$n = \frac{-b±\sqrt{b^{2}-4ac}}{2a}$  (ignore the Â, it's a formatting error)

$n = \frac{-(3)±\sqrt{(3)^{2}-4(1)(-238)}}{2(1)}$  Substitute the values for 'a', 'b' and 'c' from the equation

$n = \frac{-3±\sqrt{9-(-952)}}{2}$     In the root, two negatives make a positive

$n = \frac{-3±\sqrt{9+952}}{2}$     Add inside the root

$n = \frac{-3±\sqrt{961}}{2}$       Find the square root of 961

$n = \frac{-3±{31}{2}$     Split the equation at the ±

Split for adding:

$n = \frac{-3+31}{2}$    Simplify

n = 14

Split for subtracting:

$n = \frac{-3-31}{2}$      Simplify

n = -17

You get the answers 14 and -17 for 'n'. Decide which one is 'inadmissible', meaning it can't be right even if it solves the equation.

The problem says 'n' is a natural number. Natural numbers are all non-partial numbers count up from 1. (Non-partial means as a fraction, they are reduced to a number over 1). Since -17 is less than 1, it is inadmissible because it is not a natural number.

Therefore, 'n' is 14.