Question

if ( x = sqrt{25} ) and ( y = sqrt{4} ), then what is the value of ( (x + y)(x - y) )?
(a) 1
(b) 3
(c) 12
(d) 16
(e) 21

Explanation:

Step1: Recall the difference of squares formula

The formula for the difference of squares is \((a + b)(a - b)=a^{2}-b^{2}\). Here, \(a = x\) and \(b = y\), so \((x + y)(x - y)=x^{2}-y^{2}\).

Step2: Calculate \(x^{2}\) and \(y^{2}\)

Given \(x=\sqrt{25}\), then \(x^{2}=(\sqrt{25})^{2}\). Since \((\sqrt{a})^{2}=a\) for \(a\geq0\), we have \(x^{2} = 25\).
Given \(y = \sqrt{4}\), then \(y^{2}=(\sqrt{4})^{2}\). Using the same property \((\sqrt{a})^{2}=a\) for \(a\geq0\), we get \(y^{2}=4\).

Step3: Substitute into the formula

Substitute \(x^{2}=25\) and \(y^{2}=4\) into \(x^{2}-y^{2}\), we have \(25 - 4=21\). Wait, no, wait. Wait, \(x=\sqrt{25}=5\)? Wait, no, wait the problem says \(x = \sqrt{25}\) and \(y=\sqrt{4}\)? Wait, no, maybe there is a typo? Wait, no, the user wrote "If \(x=\sqrt{25}\) and \(y = \sqrt{4}\), then what is the value of \((x + y)(x - y)\)?" Wait, but let's re - check. Wait, \(\sqrt{25}=5\), \(\sqrt{4}=2\). Then \((x + y)(x - y)=x^{2}-y^{2}=25 - 4 = 21\)? But wait, the options have (E)21. But wait, maybe I misread \(x\). Wait, maybe \(x=\sqrt{25}\) is correct. Wait, let's do it again.
Wait, \(x=\sqrt{25}=5\), \(y = \sqrt{4}=2\). Then \((x + y)(x - y)=(5 + 2)(5 - 2)=7\times3 = 21\). But wait, the options are (A)1, (B)3, (C)12, (D)16, (E)21. So the answer should be 21. But wait, maybe there is a mistake in the problem statement? Wait, maybe \(x=\sqrt{25}\) is a typo? Wait, no, let's check the formula again. The difference of squares is \(a^{2}-b^{2}\). So \(x^{2}-y^{2}=(\sqrt{25})^{2}-(\sqrt{4})^{2}=25 - 4 = 21\).

Answer:

E. 21