High School

Use the identities to calculate the following:

[tex]\((a \pm b)^2 = a^2 \pm 2ab + b^2\)[/tex]

a) Calculate [tex]\(501^2\)[/tex]:

[tex]\[
\begin{align*}
(a+b)^2 & = a^2 + 2ab + b^2 \\
a & = 500, \quad b = 1 \\
500^2 & = 250000 \\
2 \times 500 \times 1 & = 1000 \\
1^2 & = 1 \\
250000 + 1000 + 1 & = 251001 \\
501^2 & = 251001
\end{align*}
\][/tex]

b) Calculate [tex]\(498^2\)[/tex]:

[tex]\[
\begin{align*}
(a-b)^2 & = a^2 - 2ab + b^2 \\
a & = 500, \quad b = 2 \\
500^2 & = 250000 \\
2 \times 500 \times 2 & = 2000 \\
2^2 & = 4 \\
250000 - 2000 + 4 & = 248004 \\
498^2 & = 248004
\end{align*}
\][/tex]

Answer :

To find [tex]\( 498^2 \)[/tex] using the identity [tex]\((a \pm b)^2 = a^2 \pm 2ab + b^2\)[/tex], we can follow these steps:

1. Identify [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:

Let's choose [tex]\( a = 500 \)[/tex] and [tex]\( b = -2 \)[/tex] because [tex]\( 498 \)[/tex] can be expressed as [tex]\( 500 - 2 \)[/tex].

2. Calculate [tex]\( a^2 \)[/tex], [tex]\( 2ab \)[/tex], and [tex]\( b^2 \)[/tex]:

[tex]\[
a^2 = 500^2 = 250000
\][/tex]

[tex]\[
2ab = 2 \times 500 \times (-2) = -2000
\][/tex]

[tex]\[
b^2 = (-2)^2 = 4
\][/tex]

3. Combine these results using the identity:

[tex]\[
(a - b)^2 = a^2 + 2ab + b^2
\][/tex]

Substitute the values:

[tex]\[
250000 - 2000 + 4 = 248004
\][/tex]

Therefore, [tex]\( 498^2 = 248004 \)[/tex].

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