Word Problem Leading to Quadratic Equation

  A two-digit number is such that its tens digit is greater than its unit by 2. The

two-digit number is more than 2 times the product of its digits by 23. Find the number

Answer

Let the two-digit number be = xy

first statement x = (y+2)

then xy = (y+2)y

the product of the two-digit number = y(y+2)

10(y+2) +y = 2y(y+2)+23

10y +20+y = 2y² +4y +23

11y +20 = 2y² + 4y +23

2y² +4y +23 – 11y -20

 2y² – 7y +3 =0

2y² -6y-y +3= 0

2y(y-3) -1(y-3) =0

(2y-1)(y-3)

Y =1/2 or 3

X =  (3+2) = 5

Xy =53

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