<p>The benevolent "algebraicians" have kindly bestowed upon us some slick shortcuts, known as "identities." </p>
<p>Identity 1: <strong>$ax + bx = x(a + b)$</strong></p>
<ul>
<li>$3x + 5x = x(3 + 5) = 8x$</li>
<li>Factoring out the common factor is like gathering all the x’s into one party.</li>
</ul>
<p>Identity 2: <strong>$ax - bx = x(a - b)$</strong></p>
<ul>
<li>$4x - 2x = x(4 - 2) = 2x$</li>
<li>Same party trick, but with a bit of subtraction drama.</li>
</ul>
<p>Identity 3: <strong>$(a + b)^2 = a^2 + 2ab + b^2$</strong></p>
<ul>
<li>$(x + 3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9$</li>
<li>The square of a binomial: turning a sum into a squared spectacle.</li>
</ul>
<p>Identity 4: <strong>$(a - b)^2 = a^2 - 2ab + b^2$</strong></p>
<ol>
<li>$(x - 4)^2 = x^2 - 2(x)(4) + 4^2 = x^2 - 8x + 16$</li>
<li>The square of a difference: because subtraction deserves some flair too.</li>
</ol>
<p>Identity 5: <strong>$(a + b)(a - b) = a^2 - b^2$</strong></p>
<ul>
<li>$(x + 5)(x - 5) = x^2 - 5^2 = x^2 - 25$</li>
<li>The difference of squares: when opposites attract and then subtract.</li>
</ul>
<p>These identities are your algebraic cheat codes, learn them well and use them wisely!</p>