Step 1
Let it be:
• x: The smaller number.
,
• x + 22: The larger number.
The product of both numbers is:
[tex]x(x+22)[/tex]
Since the product of both numbers is -121, we can write:
[tex]x(x+22)=-121[/tex]
The word 'is' is represented by the symbol =.
Therefore, the equation to solve the given word problem is:
[tex]x(x+22)=-121[/tex]Step 2
To find the numbers, we solve the previous equation.
[tex]\begin{gathered} x(x+22)=-121 \\ \text{ Apply the distributive property on the left side} \\ x\cdot x+22\cdot x=-121 \\ x^2+22x=-121 \\ \text{ Add }121\text{ from both sides} \\ x^2+22x+121=-121+121 \\ x^2+22x+121=0 \end{gathered}[/tex]
Now, we can factor the expression on the left side using the perfect square trinomial rule.
[tex]a^2+2ab+b^2=(a+b)^2[/tex]
Then, we have:
[tex]\begin{gathered} x^2+22x+121=0 \\ x^2+2\cdot11\cdot x+11^2=0 \\ (x+11)^2=0 \\ \text{ Apply square root to both sides of the equation} \\ \sqrt[]{(x+11)^2}=\sqrt[]{0} \\ x+11=0 \\ \text{ Subtract 11 from both sides} \\ x+11-11=0-11 \\ x=-11 \end{gathered}[/tex]
Finally, we find the another number.
[tex]\begin{gathered} x+22=-11+22 \\ x+22=11 \end{gathered}[/tex]
Therefore, the numbers are -11 and 11.
