Question

If four is added to two times a number, the result is fifty-nine less than the square of the number. Find all such numbers.

If there is more than one answer, separate them with the "and" button.
If there is no such number, select the "None" button.

Answer 1

9 and -7

Explanation:

First, translate the word problem into an algebraic equation.

Let x represent the number to be determined

4 added to 2 times a number ==> 2x + 4

59 less than square of number ==> x² - 59

Equating the two we get
x² - 59 = 2x + 4

Move the term on the right to the left; the signs will change for each term
x² - 59 - 2x - 4 = 0

x² - 2x -59 -4 = 0

x² -2x -63 = 0

This is a quadratic equation when can be solved using factoring

Factors of 63 are
-63 = -7 x 9
-63 = 7 x -9

The second factorization 7 and -9 will also yield -2 when they are added

So

x² -2x -63 = 0

x² + 7x - 9x -63 = 0

x(x + 7) -9(x + 7) = 0

Factor out common term x + 7 to get

(x + 7)(x - 9) = 0

This means
x + 7 = 0 ==> x = -7
x - 9 = 0 ==> x = 9

The solutions are therefore x = 9 and x = -7

Therefore the number can be 9 or -7