Answer:
The two integers are 18 and 3.
Step-by-step explanation:
Consider the provided information.
It is given that One integer is 3 more than 5 times another.
Let us consider the integer is <em>x</em>.
Then 3 more than 5 times of x is:
![3+5x](https://tex.z-dn.net/?f=3%2B5x)
The product of the provided two numbers is 54. i.e.
![x(3+5x)=54](https://tex.z-dn.net/?f=x%283%2B5x%29%3D54)
![3x+5x^2=54](https://tex.z-dn.net/?f=3x%2B5x%5E2%3D54)
![5x^2+3x-54=0](https://tex.z-dn.net/?f=5x%5E2%2B3x-54%3D0)
Use the formula of the quadratic equation:
For the equation
the solutions are:
![x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=x_%7B1%2C%5C%3A2%7D%3D%5Cfrac%7B-b%5Cpm%20%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D)
By comparing the obtained equation with the above equation, it can be concluded that <em>a</em> = 5, <em>b</em> = 3 and <em>c</em> = -54. Substitute the these values in the above formula.
![x_{1,\:2}=\frac{-3\pm \sqrt{3^2-4\cdot \:5\left(-54\right)}}{2\cdot \:5}](https://tex.z-dn.net/?f=x_%7B1%2C%5C%3A2%7D%3D%5Cfrac%7B-3%5Cpm%20%5Csqrt%7B3%5E2-4%5Ccdot%20%5C%3A5%5Cleft%28-54%5Cright%29%7D%7D%7B2%5Ccdot%20%5C%3A5%7D)
![x_{1,\:2}=\frac{-3\pm\sqrt{3^2+4\cdot \:5\cdot \:54}}{2\cdot \:5}](https://tex.z-dn.net/?f=x_%7B1%2C%5C%3A2%7D%3D%5Cfrac%7B-3%5Cpm%5Csqrt%7B3%5E2%2B4%5Ccdot%20%5C%3A5%5Ccdot%20%5C%3A54%7D%7D%7B2%5Ccdot%20%5C%3A5%7D)
![x_{1,\:2}=\frac{-3\pm\sqrt{1089}}{10}](https://tex.z-dn.net/?f=x_%7B1%2C%5C%3A2%7D%3D%5Cfrac%7B-3%5Cpm%5Csqrt%7B1089%7D%7D%7B10%7D)
![x_{1,\:2}=\frac{-3\pm33}{10}](https://tex.z-dn.net/?f=x_%7B1%2C%5C%3A2%7D%3D%5Cfrac%7B-3%5Cpm33%7D%7B10%7D)
or ![x_2=\frac{-3-33}{10}](https://tex.z-dn.net/?f=x_2%3D%5Cfrac%7B-3-33%7D%7B10%7D)
or ![x_2=\frac{-36}{10}](https://tex.z-dn.net/?f=x_2%3D%5Cfrac%7B-36%7D%7B10%7D)
or ![x_2=\frac{-18}{5}](https://tex.z-dn.net/?f=x_2%3D%5Cfrac%7B-18%7D%7B5%7D)
As we have given that numbers are integer, therefore ignore ![x_2=\frac{-18}{5}](https://tex.z-dn.net/?f=x_2%3D%5Cfrac%7B-18%7D%7B5%7D)
Thus, the value of <em>x</em> is 3 or one number is 3.
Then 3 more than 5 times of x is: 3 + 5(3) = 18.
Therefore, the two integers are 18 and 3.