Chapter 9.2, Problem 34PPS

To determine

The two numbers that have a sum of 9 and a product of 20

Answer to Problem 34PPS

The numbers are $x=5,4$

Explanation of Solution

Given:

Two numbers that have a sum of 9 and a product of 20

Concept Used:

• To get rid of a number in addition from one side, subtract the same number from both sides of equal sign.
• To get rid of a number in subtraction from one side, add the same number both sides of equal sign.
• To get rid of a number in multiplication from one side, divide the same number from both sides of equal sign.
• To get rid of a number in division from one side, multiply the same number both sides of equal sign.

Rules of Addition/ Subtraction:

• Two numbers with similar sign always get added and the resulting number will carry the similar sign.
• Two numbers with opposite signs always get subtracted and the resulting number will carry the sign of larger number.

Rules of Multiplication/ Division:

• The product/quotient of two similar sign numbers is always positive.
• The product/quotient of two numbers with opposite signs is always negative.

In order find the two numbers that have a sum of 9 and a product of 20. Let us assume that the first number is x than the second number will be $9-x$

Here to find the numbers shown below,

  $\begin{array}{c}x+\left(9-x\right)=9\mathrm{......}\left(i\right)\\ \text{x}\left(9-x\right)=20\mathrm{......}\left(ii\right)\\ \text{from }\left(\text{ii}\right)\text{ we get}\\ x\left(9-x\right)=20\\ 9x-x^2=20\\ -x^2+9x-20=20-20\\ -\left(-x^2+9x-20\right)=0\\ x^2-9x+20=0\\ x^2-\left(5+4\right)x+20=0\\ x^2-5x-4x+20=0\\ x\left(x-5\right)-4\left(x-5\right)=0\\ \left(x-5\right)\left(x-4\right)=0\\ \Rightarrow x=5,4\end{array}$

Thus, the numbers are $x=5,4$