Question

Two consecutive numbers can be represented by x and x+1. The product of two consecutive numbers is equal to 35 more than eleven times the sum of the numbers. x(x+1)=11(x+x+1)+35a. Write a quadratic equation to represent the situation.b. What are the solutions to the equation?c. What are the two consecutive numbers?

Answer 1

Part a

we have that

the equation that represents this problem is

x(x+1)=11(x+x+1)+35

Distribute and combine like terms

x^2+x=11(2x+1)+35

x^2+x=22x+11+35

x^2+x=22x+46

x^2+x-22x-46=0

x^2-21x-46=0

The answer Part a is

x^2-21x-46=0

Part b

Solve the quadratic equation

x^2-21x-46=0

using the formula

a=1

b=-21

c=-46

substitute

$x=\frac{-(-21)\pm\sqrt[]{-21^2-4(1)(-46)}}{2(1)}$$\begin{gathered} x=\frac{21\pm\sqrt[]{625}}{2} \\ \\ x=\frac{21\pm25}{2} \end{gathered}$

The solutions for x are

x=-2 and x=23

Part c

Two consecutive numbers

For x=-2 ------> the numbers are -2 and -1

For x=23 -----> the numbers are 23 and 24