Question

question 7 of 10
which of the following expressions is equal to ( x^2 + 25 )?

a. ( (x - 5i)^2 )
b. ( (x + 10i)(x - 15i) )
c. ( (x - 5i)(x + 5i) )
d. ( (x + 5i)^2 )

Explanation:

Step1: Recall the difference of squares formula

The difference of squares formula is \(a^2 - b^2=(a - b)(a + b)\). Also, recall that \(i^2=- 1\).

Step2: Analyze option A

Expand \((x - 5i)^2\) using the formula \((a - b)^2=a^2-2ab + b^2\). Here \(a = x\), \(b = 5i\).
\((x - 5i)^2=x^2-2\times x\times5i+(5i)^2=x^2-10xi + 25i^2\)
Since \(i^2=-1\), we have \(x^2-10xi+25\times(- 1)=x^2-10xi - 25\), which is not equal to \(x^2 + 25\).

Step3: Analyze option B

Expand \((x + 10i)(x - 15i)\) using the distributive property (FOIL method):
\(x\times x+x\times(-15i)+10i\times x+10i\times(-15i)=x^2-15xi + 10xi-150i^2\)
Combine like terms: \(x^2-5xi-150\times(-1)=x^2-5xi + 150\), which is not equal to \(x^2 + 25\).

Step4: Analyze option C

Expand \((x - 5i)(x + 5i)\) using the difference of squares formula \(a^2 - b^2=(a - b)(a + b)\), where \(a=x\) and \(b = 5i\).
\((x - 5i)(x + 5i)=x^2-(5i)^2=x^2-25i^2\)
Since \(i^2=-1\), we have \(x^2-25\times(-1)=x^2 + 25\), which is equal to the given expression.

Step5: Analyze option D (for completeness)

Expand \((x + 5i)^2\) using the formula \((a + b)^2=a^2+2ab + b^2\). Here \(a = x\), \(b = 5i\).
\((x + 5i)^2=x^2+2\times x\times5i+(5i)^2=x^2 + 10xi+25i^2\)
Since \(i^2=-1\), we have \(x^2 + 10xi+25\times(-1)=x^2 + 10xi-25\), which is not equal to \(x^2 + 25\).

Answer:

C. \((x - 5i)(x + 5i)\)